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Phase Transformations

1. ادخ منا به

2. Gernot Kostorz (Editor)
Phase Transformations in Materials
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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4. Gernot Kostorz (Ed.)
PhaseTransformations
inMaterials
Weinheim · New York · Chichester · Brisbane · Singapore · Torontowww.iran-mavad.com
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5. Editor:
Prof. Gernot Kostorz
ETH Zürich
Institut für Angewandte Physik
CH-8093 Zürich
Switzerland
This book was carefully produced. Nevertheless, authors, editor and publisher do not warrant the informa-
tion contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illus-
trations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No. applied for
British Library Cataloguing-in-Publication Data:
A catalogue record for this book is available from the British Library.
Die Deutsche Bibliothek – Cataloguing-in-Publication Data:
A catalogue record for this book is available from Die Deutsche Bibliothek
ISBN 3-527-30256-5
© WILEY-VCH Verlag GmbH, D-69469 Weinheim (Federal Republic of Germany), 2001
All rights reserved (including those of translation in other languages). No part of this book may be reproduced
in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into machine lan-
guage without written permission from the publishers. Registered names, trademarks, etc. used in this book, even
when not specifically marked as such, are not to be considered unprotected by law.
Printed in the Federal Republic of Germany
Printed on acid-free paper.
Indexing: Borkowski & Borkowski, Schauernheim
Composition, Printing and Bookbinding: Konrad Triltsch, Print und digitale Medien GmbH, 97199 Ochsenfurt-
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6. To the memory of Peter Haasen
(1927–1993)
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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7. Preface
In 1991, the late Peter Haasen, who had set out to edit a comprehensive treatment of “Ma-
terials Science and Technology” together with Robert W. Cahn and Edward J. Kramer, a
very successful series of up-to-date coverages of a broad range of materials topics, wrote
the following in a Preface to Volume 5.
“Herewith we proudly present the first volume of this Series, the aim of which is to pro-
vide a comprehensive treatment of materials science and technology. The term ‘materials’
encompasses metals, ceramics, electronic and magnetic materials, polymers and compos-
ites. In many cases these materials have been developed independently within different dis-
ciplines but are now finding uses in similar technologies. Moreover, similarities found among
the principles underlying these various disciplines have led to the discovery of common
phenomena and mechanisms. One of these common features, phase transformations, con-
stitutes the topic of this volume and rates among the fundamental phenomena central to the
Series. A phase transformation often delivers a material into its technologically useful form
and microstructure. For example, the major application of metals and alloys as mechanical-
ly strong materials relies on their multi-phase microstructure, most commonly generated by
one or more phase transformations.”
Peter Haasen, who passed away in 1993, did not live to see the overwhelming success of
this volume. A revised edition was planned as early as 1996 and finally, work on the indi-
vidual chapters began in 1998. Almost all of the original authors agreed to update their ear-
lier work, in many cases they arranged for the participation of younger colleagues who had
made major contributions to the field. It is thus with similar pride the present editor sub-
mits the second edition of “Phase Transformations” to the scholarly public. It was with some
hesitation that he assumed the task of editor, as he did not like giving the impression of pla-
giarizing a successful work. All the credit for the idea, the original chapter definitions and
the selection of the initial authors remains with Peter Haasen. He would certainly have liked
to work on a new edition himself and, assuming that he would have judged it timely to ac-
complish it about ten years after the first edition, this editor tried to help reaching this goal
in the original spirit.
Thus, all the chapters kept their original titles. The contents have been thoroughly re-
edited and updated and reflect the progress in the field up to about the middle of the year
2000. As before, the book starts with the foundations of phase transformations (“Thermo-
dynamics and Phase Diagrams” by A. D. Pelton). The sequence of the following Chapters
has been slightly modified. As most of the volume concerns the solid state, Chapter 2 (by
H. Müller-Krumbhaar, W. Kurz and E. Brener) is devoted to solidification, a subject of great
basic and technological relevance. Chapter 3 by G. E. Murch covers the most important
ideas and methods of diffusion kinetics in solids, an indispensible ingredient to many phase
transformations. Statistical theories of phase transformations are presented by K. Binder in
Chapter 4, featuring phenomenological concepts and computational methods. Diffusion con-
trolled homogeneous phase transformations are treated in Chapter 5 (“Homogeneous Sec-
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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8. ond Phase Precipitation” by R. Wagner, R. Kampmann and P. W. Voorhees) and Chapter 6
(“Spinodal Decomposition” by K. Binder and P. Fratzl), looking at the very complex kinet-
ical aspects of the formation of new phases from the points of view of metastability and in-
stability in an initially homogeneous system. Heterogeneous phase transformations are treat-
ed in Chapter 7 (“Transformations Involving Interfacial Diffusion” by G. R. Purdy and Y.
Bréchet) while Chapter 8 by G. Inden deals with atomic ordering, mostly involving substi-
tutional alloys and intermetallic phases. Though much progress has been made in elucidat-
ing equilibrium ordered states, kinetical aspects are still widely unexplored in this field. Fi-
nally, the numerous aspects of diffusionless transformations in the solid state are taken up
by L. Delaey (Chapter 9), and a completely new Chapter on the effects of pressure on phase
transformations has been provided by M. Kunz (Chapter 10).
In working on this new edition, the editor had great pleasure interacting with the authors,
those of the first edition as well as those who joined for the new edition. He is grateful to
all of them for their friendly and competent co-operation. Thanks are due to the publisher
for expedient support and for the preparation of the subject index.
It is hoped that this book will be useful as a source of reference to active researchers and
advanced students; more up-to-date and more detailed than encyclopedic articles, but not
as complete and extensive as any monographs. Phase transformations are among the most
complex and most versatile phenomena in solid state physics and materials science – and
have considerable impact on production and processing technology. The present book should
encourage the reader to enter and more deeply appreciate this challenging field.
Gernot Kostorz, Zürich
April 2001
Preface VIIwww.iran-mavad.com
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9. Prof. Kurt Binder
Universität Mainz
Institut für Physik
Staudinger Weg 7
D-55099 Mainz
Germany
Chapters 4 and 6
Prof. Yves J. M. Bréchet
Laboratoires de Thermodynamiques
at Physico-Chimique Métallurgiques BP75
Domaine Universitaire de Grenoble
F-38402 Saint Martin d’Heres
France
Chapter 7
Dr. Efim Brener
Institut für Festkörperforschung
Forschungszentrum Jülich
D-52425 Jülich
Germany
Chapter 2
Prof. Luc Delaey
Katholieke Universiteit Leuven
Dept. Metaalkunde en Toegepaste
Materiaalkunde
Decroylaan 2
B-3030 Heverlee-Leuven
Belgium
Chapter 9
Prof. Peter Fratzl
Institut für Metallphysik
Montan-Universität Leoben
Jahnstraße 12
A-8700 Leoben
Austria
Chapter 6
Prof. Gerhard Inden
Max-Planck-Institut für Eisenhütten-
forschung GmbH
Max-Planck-Str. 1
D-40237 Düsseldorf
Germany
Chapter 8
Dr. Reinhard Kampmann
GKSS-Forschungszentrum Geesthacht
GmbH
Institut für Werkstoffforschung
Postfach 1160
D-21494 Geesthacht
Germany
Chapter 5
Prof. Martin Kunz
ETH Zürich
Labor für Kristallographie
Sonneggstr. 5
CH-8092 Zürich
Switzerland
now at:
Naturhistorisches Museum Basel
Augustinerstr. 2
CH-4053 Basel
Switzerland
Chapter 10
Prof. Wilfried Kurz
École Polytechnique de Lausanne
DMX-G, Ecublens
CH-1015 Lausanne
Switzerland
Chapter 2
List of Contributors
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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10. List of Contributors IX
Prof. Gary R. Purdy
Dept. of Materials Science and Engineering
McMaster University
1280 Main Street
Hamilton, Ontario L8S 4L7
Canada
Chapter 7
Prof. Peter W. Voorhees
Dept. of Materials Science and Engineering
Northwestern University
2225 N. Campus Drive
Evanston, IL 60208-3108
USA
Chapter 5
Prof. Richard Wagner
Forschungszentrum Jülich
D-52425 Jülich
Germany
Chapter 5
Prof. Heiner Müller-Krumbhaar
Institut für Festkörperforschung
Forschungszentrum Jülich
D-52425 Jülich
Germany
Chapter 2
Prof. Graeme E. Murch
University of Newcastle
Dept. of Chemical and Materials
Engineering
Romkin Drive
Newcastle, NSW 2308
Australia
Chapter 3
Prof. Arthur D. Pelton
École Polytechnique de Montréal
Centre de Recherche en Calcul
Thermochimique
CP 6079
Succursale A
Montréal, Québec H3C 3A7
Canada
Chapter 1www.iran-mavad.com
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11. Contents
1 Thermodynamics and Phase Diagrams of Materials . . . . . . . . . . . . 1
A. D. Pelton
2 Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
H. Müller-Krumbhaar, W. Kurz, E. Brener
3 Diffusion in Crystalline Solids . . . . . . . . . . . . . . . . . . . . . . . 171
G. E. Murch
4 Statistical Theories of Phase Transitions . . . . . . . . . . . . . . . . . . 239
K. Binder
5 Homogeneous Second Phase Precipitation . . . . . . . . . . . . . . . . . 309
R. Wagner, R. Kampmann, P. W. Voorhees
6 Spinodal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
K. Binder, P. Fratzl
7 Transformations Involving Interfacial Diffusion . . . . . . . . . . . . . . . 481
G. R. Purdy, Y. J. M. Bréchet
8 Atomic Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
G. Inden
9 Diffusionless Transformations . . . . . . . . . . . . . . . . . . . . . . . . 583
L. Delaey
10 High Pressure Phase Transformations . . . . . . . . . . . . . . . . . . . . 655
M. Kunz
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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12. 1 Thermodynamics and Phase Diagrams of Materials
Arthur D. Pelton
Centre de Recherche en Calcul Thermochimique, École Polytechnique,
Montréal, Québec, Canada
List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Introduction................................. 5
1.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Gibbs Energy and Equilibrium....................... 5
1.2.1 Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Predominance Diagrams.......................... 7
1.3.1 Calculation of Predominance Diagrams . . . . . . . . . . . . . . . . . . . 7
1.3.2 Ellingham Diagrams as Predominance Diagrams . . . . . . . . . . . . . . 8
1.3.3 Discussion of Predominance Diagrams . . . . . . . . . . . . . . . . . . . 9
1.4 Thermodynamics of Solutions....................... 9
1.4.1 Gibbs Energy of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.2 Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.3 Tangent Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.4 Gibbs-Duhem Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.5 Relative Partial Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.6 Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.7 Ideal Raoultian Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.8 Excess Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.9 Activity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.10 Multicomponent Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Binary Phase Diagrams........................... 14
1.5.1 Systems with Complete Solid and Liquid Miscibility . . . . . . . . . . . . 14
1.5.2 Thermodynamic Origin of Phase Diagrams . . . . . . . . . . . . . . . . . 16
1.5.3 Pressure-Composition Phase Diagrams . . . . . . . . . . . . . . . . . . . 19
1.5.4 Minima and Maxima in Two-Phase Regions . . . . . . . . . . . . . . . . . 20
1.5.5 Miscibility Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5.6 Simple Eutectic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.7 Regular Solution Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.8 Thermodynamic Origin of Simple Phase Diagrams Illustrated by Regular
Solution Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.9 Immiscibility – Monotectics . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5.10 Intermediate Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5.11 Limited Mutual Solubility – Ideal Henrian Solutions . . . . . . . . . . . . 29
1.5.12 Geometry of Binary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . 31
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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13. 1.6 Application of Thermodynamics to Phase Diagram Analysis...... 34
1.6.1 Thermodynamic/Phase Diagram Optimization . . . . . . . . . . . . . . . . 34
1.6.2 Polynomial Representation of Excess Properties . . . . . . . . . . . . . . . 34
1.6.3 Least-Squares Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.6.4 Calculation of Metastable Phase Boundaries . . . . . . . . . . . . . . . . . 39
1.7 Ternary and Multicomponent Phase Diagrams.............. 39
1.7.1 The Ternary Composition Triangle . . . . . . . . . . . . . . . . . . . . . . 39
1.7.2 Ternary Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.7.3 Polythermal Projections of Liquidus Surfaces . . . . . . . . . . . . . . . . 41
1.7.4 Ternary Isothermal Sections . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.7.4.1 Topology of Ternary Isothermal Sections . . . . . . . . . . . . . . . . . . 45
1.7.5 Ternary Isopleths (Constant Composition Sections) . . . . . . . . . . . . . 46
1.7.5.1 Quasi-Binary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 47
1.7.6 Multicomponent Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . 47
1.7.7 Nomenclature for Invariant Reactions . . . . . . . . . . . . . . . . . . . . 49
1.7.8 Reciprocal Ternary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . 49
1.8 Phase Diagrams with Potentials as Axes.................. 51
1.9 General Phase Diagram Geometry..................... 56
1.9.1 General Geometrical Rules for All True Phase Diagram Sections . . . . . . 56
1.9.1.1 Zero Phase Fraction Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.9.2 Choice of Axes and Constants of True Phase Diagrams . . . . . . . . . . . 58
1.9.2.1 Tie-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.9.2.2 Corresponding Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . 60
1.9.2.3 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.9.2.4 Other Sets of Conjugate Pairs . . . . . . . . . . . . . . . . . . . . . . . . 61
1.10 Solution Models............................... 62
1.10.1 Sublattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.10.1.1 All Sublattices Except One Occupied by Only One Species . . . . . . . . . 62
1.10.1.2 Ionic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.10.1.3 Interstitial Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.10.1.4 Ceramic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.10.1.5 The Compound Energy Formalism . . . . . . . . . . . . . . . . . . . . . . 65
1.10.1.6 Non-Stoichiometric Compounds . . . . . . . . . . . . . . . . . . . . . . . 65
1.10.2 Polymer Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1.10.3 Calculation of Limiting Slopes of Phase Boundaries . . . . . . . . . . . . 66
1.10.4 Short-Range Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
1.10.5 Long-Range Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1.11 Calculation of Ternary Phase Diagrams From Binary Data....... 72
1.12 Minimization of Gibbs Energy....................... 74
1.12.1 Phase Diagram Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 76
1.13 Bibliography................................. 76
1.13.1 Phase Diagram Compilations . . . . . . . . . . . . . . . . . . . . . . . . . 76
1.13.2 Thermodynamic Compilations . . . . . . . . . . . . . . . . . . . . . . . . 77
1.13.3 General Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.14 References.................................. 78
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14. List of Symbols and Abbreviations 3
List of Symbols and Abbreviations
Symbol Designation
a
i activity of component i
C number of components
c
p molar heat capacity
E electrical potential of a galvanic cell
F degrees of freedom/variance
G Gibbs energy in J
g molar Gibbs energy in J/mol
g
i partial molar Gibbs energy of i
G
i
0 standard Gibbs energy of i
g
i
0 standard molar Gibbs energy of i
Dg
i relative partial Gibbs energy i
g
E
excess molar Gibbs energy
g
i
E excess partial Gibbs energy of i
DG Gibbs energy change
DG
0
standard Gibbs energy change
Dg
m molar Gibbs energy of mixing
Dg
f
0 standard molar Gibbs energy of fusion
Dg
v
0 standard molar Gibbs energy of vaporization
H enthalpy in J
h molar enthalpy in J/mol
h
i partial enthalpy of i
H
i
0 standard enthalpy of i
h
i
0 standard molar enthalpy of i
Dh
i relative partial enthalpy of i
h
E
excess molar enthalpy
h
i
E excess partial enthalpy of i
DH enthalpy change
DH
0
standard enthalpy change
Dh
m molar enthalpy of mixing
Dh
f
0 standard molar enthalpy of fusion
Dh
v
0 standard molar enthalpy of vaporization
K equilibrium constant
k
B Boltzmann constant
n number of moles
n
i number of moles of constituent i
N
i number of particles of i
N
0
Avogadro’s number
p
i partial pressure of i
P total pressure
P number of phases
q
i general extensive variablewww.iran-mavad.com
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15. R gas constant
S entropy in J/K
s molar entropy in J/mol K
s
i partial entropy of i
S
i
0 standard entropy of i
s
i
0 standard molar entropy of i
Ds
i
0 relative partial entropy of i
s
E
excess molar entropy
s
i
E excess partial entropy of i
DS entropy change
DS
0
standard entropy change
Ds
m molar entropy of mixing
Ds
f
0 standard molar entropy of fusion
Ds
v
0 standard molar entropy of vaporization
T temperature
T
f temperature of fusion
T
c critical temperature
T
E eutectic temperature
U internal energy
v
i molar volume of i
v
i
0 standard molar volume of i
X
i mole fraction of i
Z coordination number
g
i activity coefficient of i
e bond energy
h empirical entropy parameter
m
i chemical potential of i
n number of moles of “foreign” particles contributed by a mole of solute
x molar metal ratio
s vibrational bond entropy
f
i generalized thermodynamic potential
w empirical enthalpy parameter
b.c.c. body-centered cubic
f.c.c. face-centered cubic
h.c.p. hexagonal close-packed
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16. 1.2 Gibbs Energy and Equilibrium 5
1.1 Introduction
An understanding of thermodynamics
and phase diagrams is fundamental and es-
sential to the study of materials science. A
knowledge of the equilibrium state under a
given set of conditions is the starting point
in the description of any phenomenon or
process.
The theme of this chapter is the relation-
ship between phase diagrams and thermo-
dynamics. A phase diagram is a graphical
representation of the values of thermody-
namic variables when equilibrium is estab-
lished among the phases of a system. Mate-
rials scientists are used to thinking of phase
diagrams as plots of temperature versus com-
position. However, many other variables
such as total pressure and partial pressures
may be plotted on phase diagrams. In Sec.
1.3, for example, predominance diagrams
will be discussed, and in Sec. 1.8 chemical
potential–composition phase diagrams will
be presented. General rules regarding phase
diagram geometry are given in Sec. 1.9.
In recent years, a quantitative coupling
of thermodynamics and phase diagrams
has become possible. With the use of com-
puters, simultaneous optimizations of ther-
modynamic and phase equilibrium data can
be applied to the critical evaluation of bi-
nary and ternary systems as shown in Sec.
1.6. This approach often enables good esti-
mations to be made of the thermodynamic
properties and phase diagrams of multi-
component systems as discussed in Sec.
1.11. These estimates are based on structu-
ral models of solutions. Various models
such as the regular solution model, the sub-
lattice model, and models for interstitial
solutions, polymeric solutions, solutions of
defects, ordered solutions, etc. are dis-
cussed in Secs. 1.5 and 1.10.
The equilibrium diagram is always cal-
culated by minimization of the Gibbs en-
ergy. General computer programs are avail-
able for the minimization of the Gibbs en-
ergy in systems of any number of phases,
components and species as outlined in Sec.
1.12. When coupled to extensive databases
of the thermodynamic properties of com-
pounds and multicomponent solutions,
these provide a powerful tool in the study
of materials science.
1.1.1 Notation
Extensive thermodynamic properties are
represented by upper case symbols. For ex-
ample, G= Gibbs energy in J. Molar prop-
erties are represented by lower case sym-
bols. For example, g=G/n= molar Gibbs
energy in J/mol where nis the total number
of moles in the system.
1.2 Gibbs Energy and Equilibrium
1.2.1 Gibbs Energy
The Gibbs energy of a system is defined
in terms of its enthalpy, H, entropy, S, and
temperature, T :
G= H– TS (1-1)
A system at constant temperature and pres-
sure will approach an equilibrium state that
minimizes G .
As an example, consider the question of
whether silica fibers in an aluminum ma-
trix at 500 °C will react to form mullite,
Al
6Si
2O
13
If the reaction proceeds with the formation
of dnmoles of mullite then, from the stoi-
chiometry of the reaction, dn
Si=(9/2) dn,
dn
Al=–6dn, and dn
SiO
2
=–13/2dn. Since
the four substances are essentially immis-
cible at 500 °C, we need consider only the
13
2
6
9
2
SiO Al = Si Al Si O (1-2)
26 213++www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

17. standard molar Gibbs energies, G
i
0. The
Gibbs energy of the system then varies as:
where DG
0
is the standard Gibbs energy
change of reaction, Eq. (1-2), at 500 °C.
Since DG
0
< 0, the formation of mullite
entails a decrease in G. Hence, the reaction
will proceed spontaneously so as to mini-
mize G .
1.2.2 Chemical Equilibrium
The partial molar Gibbs energy of an
ideal gas is given by:
g
i= g
i
0+ RTlnp
i (1-4)
where g
i
0is the standard molar Gibbs en-
ergy (at 1 bar), p
iis the partial pressure in
bar, and Ris the gas constant. The second
term in Eq. (1-4) is entropic. As a gas ex-
pands at constant T, its entropy increases.
Consider a gaseous mixture of H
2, S
2
and H
2S with partial pressures p
H
2
, p
S
2
and
p
H
2S. The gases can react according to
2H
2+ S
2= 2 H
2S (1-5)
If the reaction, Eq. (1-5), proceeds to the
right with the formation of 2 dnmoles of
H
2S, then the Gibbs energy of the system
varies as:
dG/dn= 2g
H
2S– 2g
H
2
– g
S
2
= (2g
0
H
2S– 2g
0
H
2
– g
0
S
2
)
+ RT(2 lnp
H
2S– 2 ln p
H
2
– ln p
S
2
)
= DG
0
+ RTln (p
2
H
2Sp
–2
H
2
p
S
2
–1
)
= DG (1-6)
DG, which is the Gibbs energy change of
the reaction, Eq. (1-5), is thus a function of
the partial pressures. If DG< 0, then the re-
action will proceed to the right so as to
minimize G. In a closed system, as the re-
dd=
= = kJ (1-3)
Al Si O Si
0
SiO
Al
6213 2
Gn
G
/ggg
g
00
00
9
2
13
2
6 830
+−
−− D
action continues with the production of H
2S, p
H
2Swill increase while p
H
2
and p
S
2
will decrease. As a result, DGwill become
progressively less negative. Eventually an equilibrium state will be reached when dG/dn=DG=0.
For the equilibrium state, therefore:
DG
0
= – RT ln K (1-7)
= – RTln (p
2
H
2Sp
–2
H
2
p
S
2
–1
)
equilibrium
where K, the “equilibrium constant” of the
reaction, is the one unique value of the ra- tio (p
2
H
2Sp
–2
H
2
p
S
2
–1
) for which the system will
be in equilibrium at the temperature T.
If the initial partial pressures are such
that DG> 0, then the reaction, Eq. (1-5),
will proceed to the left in order to minimize Guntil the equilibrium condition of Eq.
(1-7) is attained.
As a further example, we consider the
possible precipitation of graphite from a gaseous mixture of CO and CO
2. The reac-
tion is:
2 CO = C + CO
2 (1-8)
Proceeding as above, we can write:
dG/dn= g
C+ g
CO
2
– 2g
CO
= (g
0
C
+ g
0
CO
2
– 2g
0
CO
) + RTln (p
CO
2
/p
2
CO
)
= DG
0
+ RTln (p
CO
2
/p
2
CO
) (1-9)
= DG= – RT ln K+ RTln (p
CO
2
/p
2
CO
)
If (p
CO
2
/p
2
CO
) is less than the equilibrium
constant K, then precipitation of graphite
will occur in order to decrease G.
Real situations are, of course, generally
more complex. To treat the deposition of
solid Si from a vapour of SiI
4, for example,
we must consider the formation of gaseous
I
2, I and SiI
2so that three reaction equa-
tions must be written:
SiI
4(g) = Si(sol) + 2 I
2(g) (1-10)
SiI
4(g) = SiI
2(g) + I
2(g) (1-11)
I
2(g) = 2 I(g) (1-12)
6 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

18. 1.3 Predominance Diagrams 7
The equilibrium state, however, is still that
which minimizes the total Gibbs energy of
the system. This is equivalent to satisfying
simultaneously the equilibrium constants
of the reactions, Eqs. (1-10) to (1-12), as
will be shown in Section 1.12 where this
example is discussed further.
1.3 Predominance Diagrams
1.3.1 Calculation of Predominance
Diagrams
Predominance diagrams are a particu-
larly simple type of phase diagram which
have many applications in the fields of hot
corrosion, chemical vapor deposition, etc.
Furthermore, their construction clearly il-
lustrates the principles of Gibbs energy
minimization and the Gibbs Phase Rule.
A predominance diagram for the Cu–
S–O system at 1000 K is shown in Fig.
1-1. The axes are the logarithms of the
partial pressures of SO
2and O
2in the
gas phase. The diagram is divided into
areas or domains of stability of the various
solid compounds of Cu, S and O. For ex-
ample, at point Z, where p
SO
2
=10
–2
and
p
O
2
=10
–7
bar, the stable phase is Cu
2O.
The conditions for coexistence of two and
three solid phases are indicated respectively
by the lines and triple points on the diagram.
For example, along the univariant line
(phase boundary) separating the Cu
2O and
CuSO
4domains the equilibrium constant
K=p
–2
SO
2
p
O
2
–3/2
of the following reaction is
satisfied:
Cu
2O + 2 SO
2+ –
3
2
O
2= 2 CuSO
4(1-13)
Hence, along this line:
(1-14)
log K= – 2 logp
SO
2
– –
3
2
log p
O
2
= constant
This boundary is thus a straight line with a
slope of (– 3/2)/2 = – 3/4.
In constructing predominance diagrams,
we define a “base element”, in this case Cu,
which must be present in all the condensed
phases. Let us further assume that there is
no mutual solubility among the condensed
phases.
Following the procedure of Bale et al.
(1986), we formulate a reaction for the for-
Figure 1-1.Predomi-
nance diagram. log p
SO
2
versus log p
O
2
(bar) at
1000 K for the Cu–S–O
system (Bale et al., 1986).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

19. mation of each solid phase, always from
one mole of the base element Cu, and in-
volving the gaseous species whose pres-
sures are used as the axes (SO
2and O
2in
this example):
Cu + –
1
2
O
2= CuO
DG= DG
0
+ RTln p
O
2
–1/2
(1-15)
Cu + –
1
4
O
2= –
1
2
Cu
2O
DG= DG
0
+ RTln p
O
2
–1/4
(1-16)
Cu + SO
2= CuS + O
2
DG= DG
0
+ RTln (p
O
2
p
–1
SO
2
) (1-17)
Cu + SO
2+ O
2= CuSO
4
DG= DG
0
+ RTln (p
–1
SO
2
p
–1
O
2
) (1-18)
and similarly for the formation of Cu
2S,
Cu
2SO
4and Cu
2SO
5.
The values of DG
0
are obtained from ta-
bles of thermodynamic properties. For any
given values of p
SO
2
and p
O
2
, DGfor each
formation reaction can then be calculated.
The stable compound is simply the one
with the most negative DG. If all the DG
values are positive, then pure Cu is the
stable compound.
By reformulating Eqs. (1-15) to (1-18) in
terms of, for example, S
2and O
2rather
than SO
2and O
2, a predominance diagram
with ln p
S
2
and ln p
O
2
as axes can be con-
structed. Logarithms of ratios or products
of partial pressures can also be used as
axes.
1.3.2 Ellingham Diagrams
as Predominance Diagrams
Rather than keeping the temperature
constant, we can use it as an axis. Figure
1-2 shows a diagram for the Fe–S–O sys-
tem in which RT lnp
O
2
is plotted versus T
at constant p
SO
2
= 1 bar. The diagram is of
the same topological type as Fig. 1-1.
A similar phase diagram of RTlnp
O
2
versus Tfor the Cu–O system is shown in
Fig. 1-3. For the formation reaction:
4 Cu + O
2= 2 Cu
2O (1-19)
we can write:
DG
0
= – RT ln K= RTln (p
O
2
)
equilibrium
= DH
0
– TDS
0
(1-20)
The diagonal line in Fig. 1-3 is thus a plot
of the standard Gibbs energy of formation
of Cu
2O versus T . The temperatures indi-
cated by the symbol M and
M are the melt-
ing points of Cu and Cu
2O respectively.
This line is thus simply a line taken from
the well-known Ellingham Diagramor
DG
0
vs. Tdiagram for the formation of
oxides. However, by drawing vertical lines
at the melting points of Cu and Cu
2O as
shown in Fig. 1-3, we convert the plot to a
true phase diagram. Stability domains for
Cu(sol), Cu(l), Cu
2O(sol), and Cu
2O(l)
are shown as functions of Tand of imposed
p
O
2
. The lines and triple points indicate
8 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-2.Predominance diagram. RTln p
O
2
ver-
sus Tat p
SO
2
= 1.0 bar for the Fe–S–O system.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

20. 1.4 Thermodynamics of Solutions 9
conditions of two- and three-phase equilib-
rium.
1.3.3 Discussion of Predominance
Diagrams
In this section discussion is limited to
the assumption that there is no mutual sol-
ubility among the condensed phases. The
calculation of predominance phase dia-
grams in which mutual solubility is taken
into account is treated in Sec. 1.9, where
the general geometrical rules governing
predominance diagrams and their relation-
ship to other types of phase diagrams are
discussed.
We frequently encounter predominance
diagrams with domains for solid, liquid,
and even gaseous compounds which have
been calculated as ifthe compounds were
immiscible, even though they may actually
be partially or even totally miscible. The
boundary lines are then no longer phase
boundaries, but are lines separating regions
in which one species “predominates”. The
well known E–pHor Pourbaix diagrams of
aqueous chemistry are examples of such
predominance diagrams.
Predominance diagrams may also be con-
structed when there are two or more base
elements, as discussed by Bale (1990).
Predominance diagrams have found
many applications in the fields of hot cor-
rosion, roasting of ores, chemical vapor
deposition, etc. A partial bibliography on
their construction and applications includes
Yokokowa (1999), Bale (1990), Bale et al.
(1986), Kellogg and Basu (1960), Ingra-
ham and Kellogg (1963), Pehlke (1973),
Garrels and Christ (1965), Ingraham and
Kerby (1967), Pilgrim and Ingraham
(1967), Gulbransen and Jansson (1970),
Pelton and Thompson (1975), Shatynski
(1977), Stringer and Whittle (1975), Spen-
cer and Barin (1979), Chu and Rahmel
(1979), and Harshe and Venkatachalam
(1984).
1.4 Thermodynamics of Solutions
1.4.1 Gibbs Energy of Mixing
Liquid gold and copper are completely
miscible at all compositions. The Gibbs en-
ergy of one mole of liquid solution, g
l
, at
1400 K is drawn in Fig. 1-4 as a function of
composition expressed as mole fraction,
X
Cu, of copper. Note that X
Au=1–X
Cu. The
curve of g
l
varies between the standard mo-
lar Gibbs energies of pure liquid Au and
Cu, g
0
Au
and g
0
Cu
.
Figure 1-3.Predominance diagram
(also known as a Gibbs energy-tem-
perature diagram or Ellingham dia-
gram) for the Cu–O system. Points
M and
Mrepresent the melting
points of the metal and oxide re-
spectively.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

21. The function Dg
l
m
shown on Fig. 1-4 is
called the molar Gibbs energy of mixing of
the liquid solution. It is defined as:
Dg
l
m
= g
l
– (X
Aug
0
Au
+ X
Cug
0
Cu
) (1-21)
It can be seen that D g
l
m
is the Gibbs energy
change associated with the isothermal mix-
ing of X
Aumoles of pure liquid Au and X
Cu
moles of pure liquid Cu to form one mole
of solution:
X
AuAu(l) + X
CuCu(l)
= 1 mole liquid solution (1-22)
Note that for the solution to be stable it is
necessary that Dg
l
m
be negative.
1.4.2 Chemical Potential
The partial molar Gibbs energy of com-
ponent i, g
i, also known as the chemical
potential,
m
i, is defined as:
g
i= m
i= (∂G/∂n
i)
T,P,n j
(1-23)
where Gis the Gibbs energy of the solu-
tion, n
iis the number of moles of compo-
nent i, and the derivative is taken with all
n
j(j9i) constant.
In the example of the Au–Cu binary liq-
uid solution, g
Cu=(∂G
l
/∂n
Cu)
T,P,n
Au
, where
G
l
=(n
Cu+n
Au)g
l
. That is, g
Cu, which has
units of J/mol, is the rate of change of the
Gibbs energy of a solution as Cu is added.
It can be seen that g
Cuis an intensive prop-
erty of the solution which depends upon
the composition and temperature but not
upon the total amount of solution. That is,
adding dn
Cumoles of copper to a solution
of given composition will (in the limit as
dn
CuÆ0) result in a change in Gibbs en-
ergy, dG, which is independent of the total
mass of the solution.
The reason that this property is called a
chemical potentialis illustrated by the fol-
lowing thought experiment. Imagine two
systems, I and II, at the same temperature
and separated by a membrane that permits
only the passage of copper. The chemical
potentials of copper in systems I and II
are g
I
Cu
=∂G
I
/∂n
I
Cu
and g
II
Cu
=∂G
II
/∂n
II
Cu
.
Copper is transferred across the membrane,
with dn
I
=–dn
II
. The change in the total
Gibbs energy accompanying this transfer
is then:
(1-24)
dG= d (G
I
+ G
II
) = – (g
I
Cu
– g
II
Cu
)dn
II
Cu
If g
I
Cu
>g
II
Cu
, then d (G
I
+ G
II
) is negative
when dn
II
Cu
is positive. That is, the total
Gibbs energy will be decreased by a trans-
fer of Cu from system I to system II.
Hence, Cu will be transferred spontane-
ously from a system of higher g
Cuto a sys-
tem of lower g
Cu. Therefore g
Cuis called
the chemical potential of copper.
An important principle of phase equilib-
rium can now be stated. When two or more
phases are in equilibrium, the chemical po-
tential of any component is the same in all
phases.
10 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-4.Molar Gibbs energy, g
l
, of liquid Au–
Cu alloys at constant temperature (1400 K) illustrat-
ing the tangent construction.www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

22. 1.4 Thermodynamics of Solutions 11
1.4.3 Tangent Construction
An important construction is illustrated
in Fig. 1-4. If a tangent is drawn to the
curve of g
l
at a certain composition
(X
Cu= 0.6 in Fig. 1-4), then the intercepts
of this tangent on the axes at X
Au=1 and
X
Cu= 1 are equal to g
Auand g
Curespec-
tively at this composition.
To prove this, we first consider that the
Gibbs energy of the solution at constant T
and P is a function of n
Auand n
Cu. Hence:
Eq. (1-25) can be integrated as follows:
where the integration is performed at con-
stant composition so that the intensive
properties g
Auand g
Cuare constant. This
integration can be thought of as describing
a process in which a pre-mixed solution of
constant composition is added to the sys-
tem, which initially contains no material.
Dividing Eqs. (1-26) and (1-25) by
(n
Au+n
Cu) we obtain expressions for the
molar Gibbs energy and its derivative:
g
l
= X
Aug
Au+ X
Cug
Cu (1-27)
and
dg
l
= g
AudX
Au+ g
CudX
Cu (1-28)
Since dX
Au=–dX
Cu, it can be seen that
Eqs. (1-27) and (1-28) are equivalent to the
tangent construction shown in Fig. 1-4.
These equations may also be rearranged
to give the following useful expression for
a binary system:
g
i= g+ (1 –X
i)dg/dX
i (1-29)
0
l
0
Au Au
0
Cu Cu
l
Au Au Cu Cu
l
Au Cu
d= d d
=(1-2
)
Gn n
Gn n
Gn n∫∫ ∫ +
+
gg
gg 6
d= d d
=d d (1-2)
l
l
Au
Au
l
Cu
Cu
Au Au Cu Cu
G
G
n
n
G
n
n
nn
TP,
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟ +
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
+gg 5
1.4.4 Gibbs–Duhem Equation
Differentiation of Eq. (1-27) yields:
dg
l
= (X
Audg
Au+ X
Cudg
Cu)
+ (g
AudX
Au+ g
CudX
Cu) (1-30)
Comparison with Eq. (1-28) then gives the
Gibbs–Duhem equationat constant T and P:
X
Audg
Au+ X
Cudg
Cu= 0 (1-31)
1.4.5 Relative Partial Properties
The difference between the partial Gibbs
energy g
iof a component in solution and
the partial Gibbs energy g
i
0of the same
component in a standard stateis called the
relative partial Gibbs energy(or relative
chemical potential), Dg
i. It is most usual to
choose as standard state the pure compo-
nent in the same phase at the same temper-
ature. The activity a
iof the component rel-
ative to the chosen standard state is then
defined in terms of Dg
iby the following
equation, as illustrated in Fig. 1-4.
Dg
i= g
i– g
i
0= m
i– m
i
0= RTlna
i(1-32)
Note that g
iand m
iare equivalent symbols,
as are g
i
0and m
i
0, see Eq. (1-23).
From Fig. 1-4, it can be seen that:
Dg
m= X
AuDg
Au+ X
CuDg
Cu
= RT(X
Aulna
Au+ X
Culna
Cu) (1-33)
The Gibbs energy of mixing can be di-
vided into enthalpy and entropy terms, as
can the relative partial Gibbs energies:
Dg
m= Dh
m– TDs
m (1-34)
Dg
i= Dh
i– TDs
i (1-35)
Hence, the enthalpy and entropy of mixing
may be expressed as:
Dh
m= X
AuDh
Au+ X
CuDh
Cu (1-36)
Ds
m= X
AuDs
Au+ X
CuDs
Cu (1-37)www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

23. and tangent constructions similar to that of
Fig. 1-4 can be used to relate the relative
partial enthalpies and entropiesDh
iand
Ds
ito the integral molar enthalpy of mix-
ing Dh
mand integral molar entropy of mix-
ing Ds
mrespectively.
1.4.6 Activity
The activity of a component in a solution
was defined by Eq. (1-32).
Since a
ivaries monotonically with g
iit
follows that when two or more phases are
in equilibrium the activity of any compo-
nent is the same in all phases, provided that
the activity in every phase is expressed
with respect to the same standard state.
The use of activities in calculations of
chemical equilibrium conditions is illus-
trated by the following example. A liquid
solution of Au and Cu at 1400 K with
X
Cu= 0.6 is exposed to an atmosphere in
which the oxygen partial pressure is
p
O
2
=10
–4
bar. Will Cu
2O be formed? The
reaction is:
2 Cu(l) + –
1
2
O
2(g) = Cu
2O(sol) (1-38)
where the Cu(l) is in solution. If the reac-
tion proceeds with the formation of dn
moles of Cu
2O, then 2 dnmoles of Cu are
consumed, and the Gibbs energy of the
Au–Cu solution changes by
– 2 (dG
l
/dn
Cu)dn
The total Gibbs energy then varies as:
dG/dn= g
Cu
2O– –
1
2
g
O
2
– 2 (dG
l
/dn
Cu)
= g
Cu
2O– –
1
2
g
O
2
– 2g
Cu
= (g
0
Cu
2O– –
1
2
g
0
O
2
– 2g
0
Cu
)
– –
1
2
RTlnp
O
2
– 2RTlna
Cu
= DG
0
+ RTln (p
O
2
–1/2
a
–2
Cu
)
= DG (1-39)
For the reaction, Eq. (1-38), at 1400 K,
DG
0
= – 68.35 kJ (Barin et al., 1977). The
activity of Cu in the liquid alloy at
X
Cu= 0.6 is a
Cu= 0.43 (Hultgren et al.,
1973). Substitution into Eq. (1-39) with
p
O
2
=10
–4
bar gives:
dG/dn= DG= – 50.84 kJ
Hence under these conditions the reaction
entails a decrease in the total Gibbs energy
and so the copper will be oxidized.
1.4.7 Ideal Raoultian Solutions
An ideal solution orRaoultian solution
is usually defined as one in which the ac-
tivity of a component is equal to its mole
fraction:
a
i
ideal= X
i (1-40)
(With a judicious choice of standard state,
this definition can also encompass ideal
Henrian solutions, as discussed in Sec.
1.5.11.)
However, this Raoultian definition of
ideality is generally only useful for simple
substitutional solutions. There are more
useful definitions for other types of solu-
tions such as interstitial solutions, ionic so-
lutions, solutions of defects, polymer solu-
tions, etc. That is, the most convenient def-
inition of ideality depends upon the solu-
tion model. This subject will be discussed
in Sec. 1.10. In the present section, Eq.
(1-40) for an ideal substitutional solution
will be developed with the Au–Cu solution
as example.
In the ideal substitutional solution model
it is assumed that Au and Cu atoms are
nearly alike, with nearly identical radii and
electronic structures. This being the case,
there will be no change in bonding energy
or volume upon mixing, so that the en-
thalpy of mixing is zero:
Dh
m
ideal= 0 (1-41)
12 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

24. 1.4 Thermodynamics of Solutions 13
Furthermore, and for the same reason, the
Au and Cu atoms will be randomly distrib-
uted over the lattice sites. (In the case of a
liquid solution we can think of the “lattice
sites” as the instantaneous atomic positions.)
For a random distribution of N
Au
gold atoms and N
Cucopper atoms over
(N
Au+N
Cu) sites, Boltzmann’s equation
can be used to calculate the configurational
entropy of the solution. This is the entropy
associated with the spatial distribution of
the particles:
(1-42)
S
config
= k
Bln (N
Au+N
Cu)!/N
Au!N
Cu!
where k
Bis Boltzmann’s constant. The con-
figurational entropies of pure Au and Cu
are zero. Hence the configurational entropy
of mixing, DS
config
, will be equal to S
config
.
Furthermore, because of the assumed close
similarity of Au and Cu, there will be no
non-configurational contribution to the en-
tropy of mixing. Hence, the entropy of
mixing will be equal to S
config
.
Applying Stirling’s approximation, which
states that lnN!=[(NlnN)–N] if N is
large, yields:
For one mole of solution, (N
Au+N
Cu)=N
0
,
where N
0
= Avogadro’s number. We also
note that (k
BN
0
) is equal to the ideal gas
constant R. Hence:
(1-44)
DS
m
ideal= – R(X
AulnX
Au+ X
CulnX
Cu)
Therefore, since the ideal enthalpy of mix-
ing is zero:
(1-45)
Dg
m
ideal= RT(X
AulnX
Au+ X
CulnX
Cu)
By comparing Eqs. (1-33) and (1-45) we
obtain:
Dg
i
ideal= RTlna
i
ideal= RTlnX
i (1-46)
DSS kNN
N
N
NN
N
N
NN
m
ideal config
BAu Cu
Au
Au
Au Cu
Cu
Cu
Au Cu== (1-)−+
×
+
+
+
⎛
⎝
⎜
⎞
⎠
⎟
()
ln ln
43
Hence Eq. (1-40) has been demonstrated
for an ideal substitutional solution.
1.4.8 Excess Properties
In reality, Au and Cu atoms are not iden-
tical, and so Au–Cu solutions are not per-
fectly ideal. The difference between a solu-
tion property and its value in an ideal solu-
tion is called an excess property. The ex-
cess Gibbs energy, for example, is defined
as:
g
E
= Dg
m– Dg
m
ideal (1-47)
Since the ideal enthalpy of mixing is zero,
the excess enthalpy is equal to the enthalpy
of mixing:
h
E
= Dh
m– Dh
m
ideal= Dh
m (1-48)
Hence:
g
E
= h
E
– Ts
E
= Dh
m– Ts
E
(1-49)
Excess partial properties are defined simi-
larly:
g
i
E= Dg
i– Dg
i
ideal
= RTlna
i– RTlnX
i (1-50)
s
i
E= Ds
i– Ds
i
ideal= Ds
i+ RlnX
i(1-51)
Also:
g
i
E= h
i
E– Ts
i
E
= Dh
i– Ts
i
E (1-52)
Equations analogous to Eqs. (1-33),
(1-36) and (1-37) relate the integral and
partial excess properties. For example, in
Au–Cu solutions:
g
E
= X
Aug
E
Au
+ X
Cug
E
Cu
(1-53)
s
E
= X
Aus
E
Au
+ X
Cus
E
Cu
(1-54)
Tangent constructions similar to that of
Fig. 1-4 can thus also be employed for ex-
cess properties, and an equation analogouswww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

25. to Eq. (1-29) can be written:
g
i
E= g
E
+ (1 –X
i)dg
E
/dX
i (1-55)
The Gibbs–Duhem equation, Eq. (1-31),
also applies to excess properties:
X
Audg
E
Au
+ X
Cudg
E
Cu
= 0 (1-56)
In Au–Cu alloys, g
E
is negative. That is,
Dg
mis more negative than Dg
m
idealand so
the solution is thermodynamically more
stable than an ideal solution. We say that
Au–Cu solutions exhibit negative devia-
tions from ideality. If g
E
> 0, then the solu-
tion is less stable than an ideal solution and
is said to exhibit positive deviations.
1.4.9 Activity Coefficient
The activity coefficientof a component
in a solution is defined as:
g
i= a
i/X
i (1-57)
From Eq. (1-50):
g
i
E= RTlng
i (1-58)
In an ideal solution
g
i=1 and g
i
E= 0 for
all components. If
g
i<1, then g
i
E< 0 and by
Eq. (1-50), Dg
i<Dg
i
ideal. That is, the com-
ponent iis more stable in the solution than
it would be in an ideal solution of the same
composition. If
g
i>1, then g
i
E> 0 and the
driving force for the component to enter
into solution is less than in the case of an
ideal solution.
1.4.10 Multicomponent Solutions
The equations of this section were de-
rived with a binary solution as an example.
However, the equations apply equally to
systems of any number of components. For
instance, in a solution of components
A–B–C–D …, Eq. (1-33) becomes:
Dg
m= X
ADg
A+ X
BDg
B+ X
CDg
C
+ X
DDg
D+ … (1-59)
1.5 Binary Phase Diagrams
1.5.1 Systems with Complete Solid
and Liquid Miscibility
The temperature–composition (T–X)
phase diagram of the CaO–MnO system is
shown in Fig. 1-5 (Schenck et al., 1964;
Wu, 1990). The abscissa is the composi-
14 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-5.Phase dia-
gram of the CaO–MnO
system at P=1 bar
(after Schenck et al.,
1964, and Wu, 1990).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

26. 1.5 Binary Phase Diagrams 15
tion, expressed as mole fraction of MnO,
X
MnO. Note that X
MnO=1–X
CaO. Phase di-
agrams are also often drawn with the com-
position axis expressed as weight percent.
At all compositions and temperatures in
the area above the line labelled liquidus, a
single-phase liquid solution will be ob-
served, while at all compositions and tem-
peratures below the line labelled solidus,
there will be a single-phase solid solution.
A sample at equilibrium at a temperature
and overall composition between these two
curves will consist of a mixture of solid
and liquid phases, the compositions of
which are given by the liquidus and solidus
compositions at that temperature. For ex-
ample, a sample of overall composition
X
MnO= 0.60 at T = 2200°C (at point Rin
Fig. 1-5) will consist, at equilibrium, of a
mixture of liquid of composition X
MnO=
0.70 (point Q) and solid of composition
X
MnO= 0.35 (point P).
The line PQis called a tie-lineor co-
node. As the overall composition is varied
at 2200 °C between points Pand Q, the
compositions of the solid and liquid phases
remain fixed at P and Q, and only the rela-
tive proportions of the two phases change.
From a simple mass balance, we can derive
the lever rulefor binary systems: (moles of
liquid)/(moles of solid) =PR/RQ. Hence,
at 2200 °C a sample with overall composi-
tion X
MnO= 0.60 consists of liquid and solid
phases in the molar ratio (0.60 – 0.35)/
(0.70–0.60) = 2.5. Were the composition
axis expressed as weight percent, then the
lever rule would give the weight ratio of
the two phases.
Suppose that a liquid CaO–MnO solu-
tion with composition X
MnO= 0.60 is
cooled very slowly from an initial tempera-
ture of about 2500 °C. When the tempera-
ture has decreased to the liquidus tempera-
ture 2270 °C (point B), the first solid
appears, with a composition at point A
(X
MnO= 0.28). As the temperature is de-
creased further, solid continues to precipi-
tate with the compositions of the two
phases at any temperature being given by
the liquidus and solidus compositions at
that temperature and with their relative
proportions being given by the lever rule.
Solidification is complete at 2030 °C, the
last liquid to solidify having composition
X
MnO= 0.60 (point C).
The process just described is known as
equilibrium cooling. At any temperature
during equilibrium cooling the solid phase
has a uniform (homogeneous) composition.
In the preceding example, the composition
of the solid phase during cooling varies
along the line APC. Hence, in order for the
solid grains to have a uniform composition
at any temperature, diffusion of CaO from
the center to the surface of the growing
grains must occur. Since solid-state dif-
fusion is a relatively slow process, equi-
librium cooling conditions are only ap-
proached if the temperature is decreased
very slowly. If a sample of composition
X
MnO= 0.60 is cooled very rapidly from the
liquid, concentration gradients will be ob-
served in the solid grains, with the concen-
tration of MnO increasing towards the sur-
face from a minimum of X
MnO= 0.28 (point
A) at the center. Furthermore, in this case
solidification will not be complete at
2030 °C since at 2030°C the average con-
centration of MnO in the solid particles
will be less than X
MnO= 0.60. These con-
siderations are discussed more fully in
Chapter 2 of this volume (Müller-Krumb-
haar et al., 2001).
At X
MnO= 0 and X
MnO= 1 in Fig. 1-5 the
liquidus and solidus curves meet at the
equilibrium melting points, or tempera-
tures of fusionof CaO and MnO, which are
T
0
f(CaO)
= 2572 °C, T
0
f (MnO)
= 1842 °C.
The phase diagram is influenced by the
total pressure, P. Unless otherwise stated,www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

27. T–Xdiagrams are usually presented for
P= const. = 1 bar. For equilibria involving
only solid and liquid phases, the phase
boundaries are typically shifted only by the
order of a few hundredths of a degree per
bar change in P. Hence, the effect of pres-
sure upon the phase diagram is generally
negligible unless the pressure is of the or-
der of hundreds of bars. On the other hand,
if gaseous phases are involved then the ef-
fect of pressure is very important. The ef-
fect of pressure will be discussed in Sec.
1.5.3.
1.5.2 Thermodynamic Origin
of Phase Diagrams
In this section we first consider the ther-
modynamic origin of simple “lens-shaped”
phase diagrams in binary systems with
complete liquid and solid miscibility.
An example of such a diagram was given
in Fig. 1-5. Another example is the Ge–Si
phase diagram in the lowest panel of Fig.
1-6 (Hansen, 1958). In the upper three pan-
els of Fig. 1-6, the molar Gibbs energies of
the solid and liquid phases, g
s
and g
l
, at
three temperatures are shown to scale. As
illustrated in the top panel, g
s
varies with
composition between the standard molar
Gibbs energies of pure solid Ge and of pure
solid Si, g
Ge
0 (s)and g
Si
0(s), while g
l
varies
between the standard molar Gibbs energies
of the pure liquid components g
Ge
0 (l)and
g
Si
0 (l).
The difference between g
Ge
0 (l)and g
Si
0(s)is
equal to the standard molar Gibbs energy
of fusion (melting) of pure Si, Dg
0
f (Si)
=
(g
Si
0(l)–g
Si
0(s)). Similarly, for Ge, Dg
0
f (Ge)
=
(g
Ge
0(l)–g
Ge
0(s)). The Gibbs energy of fusion of
a pure component may be written as:
Dg
f
0= Dh
f
0– TDs
f
0 (1-60)
where Dh
f
0and Ds
f
0are the standard molar
enthalpy and entropy of fusion.
Since, to a first approximation, Dh
f
0and
Ds
f
0are independent of T, Dg
f
0is approxi-
mately a linear function of T. If T >T
f
0,
then Dg
f
0is negative. If T<T
f
0, then Dg
f
0is
positive. Hence, as seen in Fig. 1-6, as T
decreases, the g
s
curve descends relative to
g
l
. At 1500 °C, g
l
<g
s
at all compositions.
Therefore, by the principle that a system
always seeks the state of minimum Gibbs
energy at constant Tand P, the liquid phase
is stable at all compositions at 1500 °C.
At 1300 °C, the curves of g
s
and g
l
cross.
The line P
1Q
1, which is the common tan-
gentto the two curves, divides the compo-
sition range into three sections. For compo-
sitions between pure Ge and P
1, a single-
phase liquid is the state of minimum Gibbs
energy. For compositions between Q
1and
pure Si, a single-phase solid solution is the
stable state. Between P
1and Q
1, a total
Gibbs energy lying on the tangent line
P
1Q
1may be realized if the system adopts
a state consisting of two phases with com-
positions at P
1and Q
1and with relative
proportions given by the lever rule. Since
the tangent line P
1Q
1lies below both g
s
and
g
l
, this two-phase state is more stable than
either phase alone. Furthermore, no other
line joining any point on g
l
to any point on
g
s
lies below the line P
1Q
1. Hence, this line
represents the true equilibrium state of the
system, and the compositions P
1and Q
1are
the liquidus and solidus compositions at
1300 °C.
As Tis decreased to 1100 °C, the points
of common tangency are displaced to
higher concentrations of Ge. For T<937°C,
g
s
<g
l
at all compositions.
It was shown in Fig. 1-4 that if a tangent
is drawn to a Gibbs energy curve, then the
intercept of this tangent on the axis at X
i=1
is equal to the partial Gibbs energy or
chemical potential g
iof component i. The
common tangent constructionof Fig. 1-6
thus ensures that the chemical potentials of
16 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
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28. 1.5 Binary Phase Diagrams 17
Ge and Si are equal in the solid and liquid
phases at equilibrium. That is:
g
l
Ge
= g
s
Ge
(1-61)
g
l
Si
= g
s
Si
(1-62)
This equality of chemical potentials was
shown in Sec. 1.4.2 to be the criterion for
phase equilibrium. That is, the common
tangent construction simultaneously mini-
mizes the total Gibbs energy and ensures
the equality of the chemical potentials,
thereby showing that these are equivalent
criteria for equilibrium between phases.
If we rearrange Eq. (1-61), subtracting
the Gibbs energy of fusion of pure Ge,
Dg
0
f (Ge)
=(g
Ge
0(l)–g
Ge
0(s)), from each side, we
get:
(g
l
Ge
– g
Ge
0(l)) – (g
s
Ge
– g
Ge
0(s))
= – (g
Ge
0 (l)– g
Ge
0(s)) (1-63)
Using Eq. (1-32), we can write Eq. (1-63)
as:
Dg
l
Ge
– Dg
s
Ge
= – Dg
0
f (Ge)
(1-64)
or
RTlna
l
Ge
– RTlna
s
Ge
= – Dg
0
f (Ge)
(1-65)
Figure 1-6.Ge–Si phase diagram at
P=1 bar (after Hansen, 1958) and
Gibbs energy composition curves at
three temperatures, illustrating the com-
mon tangent construction (reprinted
from Pelton, 1983).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

29. where a
l
Ge
is the activity of Ge (with re-
spect to pure liquid Ge as standard state) in
the liquid solution on the liquidus, and a
s
Ge
is the activity of Ge (with respect to pure
solid Ge as standard state) in the solid solu-
tion on the solidus. Starting with Eq. (1-
62), we can derive a similar expression for
the other component:
RTlna
l
Si
– RTlna
s
Si
= – Dg
0
f (Si)
(1-66)
Eqs. (1-65) and (1-66) are equivalent to the
common tangent construction.
It should be noted that absolute values of
Gibbs energies cannot be defined. Hence,
the relative positions of g
Ge
0 (l)and g
Si
0 (l)in
Fig. 1-6 are completely arbitrary. However,
this is immaterial for the preceding discus-
sion, since displacing both g
Si
0 (l)and g
Si
0(s)by
the same arbitrary amount relative to g
Ge
0 (l)
and g
Ge
0(s)will not alter the compositions of
the points of common tangency.
It should also be noted that in the present
discussion of equilibrium phase diagrams
we are assuming that the physical dimen-
sions of the single-phase regions in the
system are sufficiently large that surface
(interfacial) energy contributions to the
Gibbs energy can be neglected. For very
fine grain sizes in the sub-micron range,
however, surface energy effects can notice-
ably influence the phase boundaries.
The shape of the two-phase (solid + liq-
uid) “lens” on the phase diagram is deter-
mined by the Gibbs energies of fusion,
Dg
f
0, of the components and by the mixing
terms, Dg
s
and Dg
l
. In order to observe
how the shape is influenced by varying
Dg
f
0, let us consider a hypothetical system
A–B in which Dg
s
and Dg
l
are ideal Raoul-
tian (Eq. (1-45)). Let T
0
f(A)
=800K and
T
0
f(B)
= 1200 K. Furthermore, assume that
the entropies of fusion of A and B are equal
and temperature-independent. The enthalp-
ies of fusion are then given from Eq. (1-60)
by the expression Dh
f
0=T
f
0Ds
f
0since
Dg
f
0= 0 when T =T
f
0. Calculated phase dia-
grams for Ds
f
0= 3, 10 and 30 J/mol K are
shown in Fig. 1-7. A value of Ds
f
0≈10 is
typical of most metals. However, when the
components are ionic compounds such as
ionic oxides, halides, etc., Ds
f
0can be sig-
nificantly larger since there are several
ions per formula unit. Hence, two-phase
“lenses” in binary ionic salt or oxide phase
diagrams tend to be “fatter” than those
encountered in alloy systems. If we are
considering vapor–liquid equilibria rather
than solid–liquid equilibria, then the
shape is determined by the entropy of
vaporization, Ds
v
0. Since Ds
v
0is usually an
order of magnitude larger than D s
f
0, two-
phase (liquid + vapor) lenses tend to be
18 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-7.Phase diagram of a system A–B with
ideal solid and liquid solutions. Melting points of A
and B are 800 and 1200 K, respectively. Diagrams
are calculated for entropies of fusion ∆S
0
f(A)
=∆S
0
f(B)
=
3, 10 and 30 J/mol K.www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

30. 1.5 Binary Phase Diagrams 19
very wide. For equilibria between two solid
solutions of different crystal structure, the
shape is determined by the entropy of
solid–solid transformation, which is usu-
ally smaller than the entropy of fusion by
approximately an order of magnitude.
Therefore two-phase (solid + solid) lenses
tend to be very narrow.
1.5.3 Pressure–Composition Phase
Diagrams
Let us consider liquid–vapor equilib-
rium with complete miscibility, using as an
example the Zn–Mg system. Curves of g
v
and g
l
can be drawn at any given Tand P,
as in the upper panel of Fig. 1-8, and the
common tangent construction then gives
the equilibrium vapor and liquid composi-
tions. The phase diagram depends upon the
Gibbs energies of vaporization of the com-
ponents Dg
v(Zn)and Dg
v(Mg)as shown in
Fig. 1-8.
To generate the isothermal pressure–
composition (P –X) phase diagram in the
lower panel of Fig. 1-8 we require the
Gibbs energies of vaporization as functions
of P. Assuming monatomic ideal vapors
and assuming that pressure has negligible
effect upon the Gibbs energy of the liquid,
we can write:
Dg
v(i)= Dg
0
v(i)
+ RTlnP (1-67)
where Dg
v(i)is the standard Gibbs energy
of vaporization (when P=1 bar), which is
given by:
Dg
0
v(i)
= Dh
0
v(i)
– TDs
0
v(i)
(1-68)
For example, the enthalpy of vaporization
of Zn is Dh
0
v (Zn)
= 115 300 J/mol at its nor-
mal boiling point of 1180 K (Barin et al.,
1977). Assuming that Dh
0
v
is independent
of T, we calculate from Eq. (1-68) that
Ds
0
v (Zn)
= 115 300/1180 = 97.71 J/mol K.
From Eq. (1-67), D g
v(Zn)at any T and Pis
thus given gy:
(1-69)
Dg
v(Zn)= (115 300 – 97.71T) + RTlnP
A similar expression can be derived for the
other component Mg.
At constant temperature, then, the curve
of g
v
in Fig. 1-8 descends relative to g
l
as
the pressure is lowered, and the P–Xphase
diagram is generated by the common tan-
gent construction. The diagram at 1250 K
in Fig. 1-8 was calculated under the as-
sumption of ideal liquid and vapor mixing
(g
E(l)
=0, g
E(v)
= 0).
P–Xphase diagrams involving liquid–
solid or solid–solid equilibria can be cal-
culated in a similar fashion through the fol-
Figure 1-8.Pressure–composition phase diagram
of the Zn–Mg system at 1250 K calculated for ideal
vapor and liquid solutions. Upper panel illustrates
common tangent construction at a constant pressure.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

31. lowing general equation, which gives the
effect of pressure upon the Gibbs energy
change for the transformation of one mole
of pure component ifrom an a-phase to a
b-phase:
where Dg
0
aÆb
is the standard (P=1 bar)
Gibbs energy of transformation, and v
i
band
v
i
aare the molar volumes.
1.5.4 Minima and Maxima
in Two-Phase Regions
As discussed in Sec. 1.4.8, the Gibbs en-
ergy of mixing Dg
mmay be expressed as
the sum of an ideal term Dg
m
idealand an ex-
cess term g
E
. As has just been shown in
Sec. 1.5.2, if Dg
s
m
and Dg
l
m
for the solid
and liquid phases are both ideal, then a
“lens-shaped” two-phase region always re-
sults. However in most systems even ap-
proximately ideal behavior is the exception
rather than the rule.
Curves of g
s
and g
l
for a hypothetical
system A–B are shown schematically in
Fig. 1-9 at a constant temperature (below
the melting points of pure A and B) such
DD
ab ab
b agg vv
→ → +−∫=d (1-)
=
0
1
70
P
P
i i
P()
that the solid state is the stable state for both pure components. However, in this system g
E(l)
<g
E(s)
, so that g
s
presents a
flatter curve than does g
l
and there exists a
central composition region in which g
l
<g
s
.
Hence, there are two common tangent lines, P
1Q
1and P
2Q
2. Such a situation
gives rise to a phase diagram with a mini- mum in the two-phase region, as observed in the Na
2CO
3–K
2CO
3system (Dessu-
reault et al., 1990) shown in Fig. 1-10. At a composition and temperature correspond- ing to the minimum point, liquid and solid of the same composition exist in equilib- rium.
A two-phase region with a minimum
point as in Fig. 1-10 may be thought of as a two-phase “lens” which has been “pushed down” by virtue of the fact that the liquid is relatively more stable than the solid. Ther- modynamically, this relative stability is ex- pressed as g
E(l)
<g
E(s)
.
Conversely, if g
E(l)
>g
E(s)
to a sufficient
extent, then a two-phase region with a maximum will result. Such maxima in (liq- uid + solid) or (solid + solid) two-phase re- gions are nearly always associated with the existence of an intermediate phase, as will be discussed in Sec. 1.5.10.
20 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-9.Isothermal Gibbs energy-composition
curves for solid and liquid phases in a system A–B in
which g
E(l)
>g
E(s)
. A phase diagram of the type of
Fig. 1-10 results.
Figure 1-10.Phase diagram of the K
2CO
3–Na
2CO
3
system at P=1 bar (Dessureault et al., 1990).www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

32. 1.5 Binary Phase Diagrams 21
1.5.5 Miscibility Gaps
If g
E
> 0, then the solution is thermody-
namically less stable than an ideal solution.
This can result from a large difference in
size of the component atoms, ions or mole-
cules, which will lead to a (positive) lattice
strain energy, or from differences in elec-
tronic structure, or from other factors.
In the Au–Ni system, g
E
is positive in
the solid phase. In the top panel of Fig. 1-11,
g
E (s)
is plotted at 1200 K (Hultgren et al.,
1973) and the ideal Gibbs energy of
mixing, Dg
m
ideal, is also plotted at 1200 K.
The sum of these two terms is the Gibbs
energy of mixing of the solid solution,
Dg
m
s, which is plotted at 1200 K as well
as at other temperatures in the central panel
of Fig. 1-11. Now, from Eq. (1-45), Dg
m
ideal
is always negative and varies directly
with T, whereas g
E
varies less rapidly with
temperature. As a result, the sum Dg
m
s=
Dg
m
ideal+g
E
becomes less negative as Tde-
creases. However, the limiting slopes to the
Dg
m
idealcurve at X
Au=1 and X
Ni=1 are both
infinite, whereas the limiting slopes of g
E
are always finite (Henry’s Law). Hence,
Dg
s
m
will always be negative as X
AuÆ1
and X
NiÆ1 no matter how low the temper-
ature. As a result, below a certain tempera-
ture the curve of Dg
s
m
will exhibit two neg-
ative “humps”. Common tangent lines
P
1Q
1, P
2Q
2, P
3Q
3to the two humps at dif-
ferent temperatures define the ends of tie-
lines of a two-phase solid–solid miscibility
gapin the Au–Ni phase diagram, which is
shown in the lower panel in Fig. 1-11
(Hultgren et al., 1973). The peak of the gap
occurs at the critical or consolute tempera-
ture and composition, T
cand X
c.
When g
E (s)
is positive for the solid phase
in a system it is usually also the case that
g
E(l)
<g
E (s)
since the unfavorable factors
(such as a difference in atomic dimensions)
which are causing g
E(s)
to be positive will
have less of an effect upon g
E (l)
in the liq-
uid phase owing to the greater flexibility of
the liquid structure to accommodate differ-
ent atomic sizes, valencies, etc. Hence, a
solid–solid miscibility gap is often asso-
ciated with a minimum in the two-phase
(solid + liquid) region, as is the case in the
Au–Ni system.
Figure 1-11.Phase diagram (after Hultgren et al.,
1973) and Gibbs energy–composition curves of solid
solutions for the Au–Ni system at P=1 bar. Letters
“s” indicate spinodal points (Reprinted from Pelton,
1983).www.iran-mavad.com
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33. Below the critical temperature the curve
of Dg
s
m
exhibits two inflection points, indi-
cated by the letter “s” in Fig. 1-11. These
are known as the spinodal points. On the
phase diagram their locus traces out the
spinodal curve(Fig. 1-11). The spinodal
curve is not part of the equilibrium phase
diagram, but it is important in the kinetics
of phase separation, as discussed in Chap-
ter 6 (Binder and Fratzl, 2001).
1.5.6 Simple Eutectic Systems
The more positive g
E
is in a system, the
higher is T
cand the wider is the miscibility
gap at any temperature. Suppose that g
E(s)
is so positve that T
cis higher than the min-
imum in the (solid + liquid) region. The re-
sult will be a phase diagram such as that of
the MgO–CaO system shown in Fig. 1-12
(Doman et al., 1963; Wu, 1990).
The lower panel of Fig. 1-12 shows the
Gibbs energy curves at 2450 °C. The two
common tangents define two two-phase re-
gions. As the temperature is decreased be-
low 2450 °C, the g
s
curve descends relative
to g
l
and the two points of tangency P
1
and P
2approach each other until, at T=
2374 °C, P
1and P
2become coincident at
the composition E. That is, at T= 2374 °C
22 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-12.Phase diagram
at P=1 bar (after Doman et
al., 1963, and Wu, 1990) and
Gibbs energy–composition
curves at 2450°C for the
MgO–CaO system. Solid
MgO and CaO have the
same crystal structure.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

34. 1.5 Binary Phase Diagrams 23
there is just one common tangent line con-
tacting the two portions of the g
s
curve at
compositions A and B and contacting the g
l
curve at E. This temperature is known as
the eutectic temperature, T
E, and the com-
position E is the eutectic composition. For
temperatures below T
E, g
l
lies completely
above the common tangent to the two por-
tions of the g
s
curve and so for T<T
Ea
solid–solid miscibility gap is observed.
The phase boundaries of this two-phase re-
gion are called the solvuslines. The word
eutectic is from the Greek for “to melt
well” since the system has its lowest melt-
ing point at the eutectic composition E.
This description of the thermodynamic
origin of simple eutectic phase diagrams is
strictly correct only if the pure solid com-
ponents A and B have the same crystal
structure. Otherwise, a curve for g
s
which
is continuous at all compositions cannot be
drawn.
Suppose a liquid MgO–CaO solution of
composition X
CaO= 0.52 (composition P
1)
is cooled from the liquid state very slowly
under equilibrium conditions. At 2450 °C
the first solid appears with composition Q
1.
As Tdecreases further, solidification con-
tinues with the liquid composition follow-
ing the liquidus curve from P
1to E and the
composition of the solid phase following
the solidus curve from Q
1to A. The rela-
tive proportions of the two phases at any T
are given by the lever rule. At a tempera-
ture just above T
E, two phases are ob-
served: a solid of composition A and a liq-
uid of composition E. At a temperature just
below T
E, two solids with compositions A
and B are observed. Therefore, at T
E, dur-
ing cooling, the following binary eutectic
reaction occurs:
liquid Æsolid
1+ solid
2 (1-71)
Under equilibrium conditions the tempera-
ture will remain constant at T=T
Euntil all
the liquid has solidified, and during the re-
action the compositions of the three phases
will remain fixed at A, B and E. For this
reason the eutectic reaction is called an in-
variantreaction. More details on eutectic
solidification may be found in Chapter 2
(Müller-Krumbhaar et al., 2001).
1.5.7 Regular Solution Theory
Many years ago Van Laar (1908) showed
that the thermodynamic origin of a great
many of the observed features of binary
phase diagrams can be illustrated at least
qualitatively by simple regular solution
theory. A simple regular solutionis one for
which:
g
E
= X
AX
B(w–hT) (1-72)
where
wand hare parameters independent
of temperature and composition. Substitut-
ing Eq. (1-72) into Eq. (1-29) yields, for
the partial properties:
(1-73)
g
E
A
= X
B
2(w–hT),g
E
B
= X
A
2(w–hT)
Several liquid and solid solutions con-
form approximately to regular solution be-
havior, particularly if g
E
is small. Examples
may be found for alloys, molecular solu-
tions, and ionic solutions such as molten
salts and oxides, among others. (The very
low values of g
E
observed for gaseous solu-
tions generally conform very closely to Eq.
(1-72).)
To understand why this should be so, we
only need a very simple model. Suppose
that the atoms or molecules of the compo-
nents A and B mix substitutionally. If the
atomic (or molecular) sizes and electronic
structures of A and B are similar, then the
distribution will be nearly random, and the
configurational entropy will be nearly
ideal. That is:
g
E
≈Dh
m– TS
E (non-config)
(1-74)www.iran-mavad.com
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35. More will be said on this point in Sec.
1.10.5.
We now assume that the bond energies
e
AA, e
BBand e
ABof nearest-neighbor pairs
are independent of temperature and com-
position and that the average nearest-
neighbor coordination number, Z , is also
constant. Finally, we assume that the en-
thalpy of mixing results mainly from the
change in the total energy of nearest-neigh-
bor pair bonds.
In one mole of solution there are (N
0
Z/2) neareast-neighbor pair bonds, where
N
0
is Avogadro’s number. Since the distri-
bution is assumed random, the probability
that a given bond is an A–A bond is equal
to X
2
A
. The probabilities of B–B and A–B
bonds are, respectively, X
2
B
and 2X
AX
B.
The molar enthalpy of mixing is then equal
to the sum of the energies of the nearest-
neighbor bonds in one mole of solution,
minus the energy of the A–A bonds in X
A
moles of pure A and the energy of the B–B
bonds in X
Bmoles of pure B:
Dh
m= (N
0
Z/2)
¥(X
2
A
e
AA+ X
2
B
e
BB+ 2X
ABe
AB)
– (N
0
Z/2) (X
Ae
AA) – (N
0
Z/2) (X
Be
BB)
= (N
0
Z) [e
AB– (e
AA+ e
BB)/2] X
AX
B
= wX
AX
B (1-75)
We now define
s
AB, s
AAand s
BBas the
vibrational entropies of nearest-neighbor
pair bonds. Following an identical argu-
ment to that just presented for the bond
energies we obtain:
s
E (non-config)
(1-76)
= (N
0
Z) [s
AB– (s
AA+s
BB)/2] =hX
AX
B
Eq. (1-72) has thus been derived. If A–B
bonds are stronger than A–A and B–B
bonds, then (
e
AB–h
ABT)<[(e
AA–h
AAT)/2
+(
e
BB–h
BBT)/2]. Hence, (w–hT)<0
and g
E
< 0. That is, the solution is rendered
more stable. If the A–B bonds are rela-
tively weak, then the solution is rendered
less stable, (
w–hT) > 0 and g
E
>0.
Simple non-polar molecular solutions
and ionic solutions such as molten salts of-
ten exhibit approximately regular behavior.
The assumption of additivity of the energy
of pair bonds is probably reasonably realis-
tic for van der Waals or coulombic forces.
For alloys, the concept of a pair bond is, at
best, vague, and metallic solutions tend to
exhibit larger deviations from regular be-
havior.
In several solutions it is found that
|
hT|<|w|in Eq. (1-72). That is, g
E
≈Dh
m
=wX
AX
B, and to a first approximation
g
E
is independent of T. This is more often
the case in non-metallic solutions than in
metallic solutions.
1.5.8 Thermodynamic Origin
of Simple Phase Diagrams Illustrated
by Regular Solution Theory
Figure 1-13 shows several phase dia-
grams, calculated for a hypothetical system
A–B containing a solid and a liquid phase
with melting points of T
0
f(A)
= 800 K and
T
0
f(B)
= 1200 K and with entropies of fusion
of both A and B set to 10 J/mol K, which is
a typical value for metals. The solid and
liquid phases are both regular with temper-
ature-independent excess Gibbs energies
g
E(s)
= w
s
X
AX
Bandg
E (l)
= w
l
X
AX
B
The parameters w
s
and w
l
have been varied
systematically to generate the various pan-
els of Fig. 1-13.
In panel (n) both phases are ideal. Panels
(l) to (r) exhibit minima or maxima de-
pending upon the sign and magnitude of
(g
E(l)
–g
E(s)
), as has been discussed in Sec.
1.5.4. In panel (h) the liquid is ideal but
positive deviations in the solid give rise to
a solid–solid miscibility gap as discussed
in Sec. 1.5.6. On passing from panel (h) to
24 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
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36. 1.5 Binary Phase Diagrams 25
panel (c), an increase in g
E(s)
results in a
widening of the miscibility gap so that the
solubility of A in solid B and of B in solid
A decreases. Panels (a) to (c) illustrate that
negative deviations in the liquid cause a
relative stabilization of the liquid with re-
sultant lowering of the eutectic tempera-
ture.
Eutectic phase diagrams are often drawn
with the maximum solid solubility occur-
ring at the eutectic temperature (as in Fig.
1-12). However, panel (d) of Fig. 1-13, in
Figure 1-13.Topological changes in the phase diagram for a system A–B with regular solid and liquid phases,
brought about by systematic changes in the regular solution parameters
ω
s
and ω
l
. Melting points of pure A and
B are 800 K and 1200 K. Entropies of fusion of both A and B are 10.0 J/mol K (Pelton and Thompson, 1975).
The dashed curve in panel (d) is the metastable liquid miscibility gap (Reprinted from Pelton, 1983).www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

37. which the maximum solubility of A in the
B-rich solid solution occurs at approxi-
mately T= 950 K, illustrates that this need
not be the case even for simple regular so-
lutions.
1.5.9 Immiscibility – Monotectics
In Fig. 1-13(e), positive deviations in
the liquid have given rise to a liquid–liquid
miscibilitygap. The CaO–SiO
2system
(Wu, 1990), shown in Fig. 1-14, exhibits
such a feature. Suppose that a liquid of
composition X
SiO
2
= 0.8 is cooled slowly
from high temperatures. At T= 1815 °C the
miscibility gap boundary is crossed and a
second liquid layer appears with a compo-
sition of X
SiO
2
= 0.97. As the temperature is
lowered further, the composition of each
liquid phase follows its respective phase
boundary until, at 1692 °C, the SiO
2-rich
liquid has a composition of X
SiO
2
= 0.99
(point B), and in the CaO-rich liquid
X
SiO
2
= 0.74 (point A). At any temperature,
the relative amounts of the two phases are
given by the lever rule.
At 1692 °C the following invariant bi-
nary monotectic reactionoccurs upon cool-
ing:
Liquid B Æ Liquid A + SiO
2(solid) (1-77)
The temperature remains constant at
1692 °C and the compositions of the phases
remain constant until all of liquid B is con-
sumed. Cooling then continues with pre-
cipitation of solid SiO
2with the equilib-
rium liquid composition following the liq-
uidus from point A to the eutectic E.
Returning to Fig. 1-13, we see in panel
(d) that the positive deviations in the liquid
in this case are not large enough to produce
immiscibility, but they do result in a flat-
tening of the liquidus, which indicates a
“tendency to immiscibility”. If the nuclea-
tion of the solid phases can be suppressed
by sufficiently rapid cooling, then a meta-
stable liquid–liquid miscibility gapis ob-
served as shown in Fig. 1-13(d). For exam-
ple, in the Na
2O–SiO
2system the flattened
(or “S-shaped”) SiO
2liquidus heralds the
existence of a metastable miscibility gap of
importance in glass technology.
1.5.10 Intermediate Phases
The phase diagram of the Ag–Mg
system (Hultgren et al., 1973) is shown in
Fig. 1-15(d). An intermetallic phase, b¢, is
seen centered approximately about the
composition X
Mg= 0.5. The Gibbs energy
curve at 1050 K for such an intermetallic
phase has the form shown schematically in
Fig. 1-15(a). The curve g
b¢
rises quite rap-
idly on either side of its minimum, which
occurs near X
Mg= 0.5. As a result, the b¢
phase appears on the phase diagram only
over a limited composition range. This
form of the curve g
b¢
results from the fact
that when X
Ag≈X
Mga particularly stable
crystal structure exists in which Ag and Mg
atoms preferentially occupy different sites.
The two common tangents P
1Q
1and P
2Q
2
give rise to a maximum in the two-phase
(b¢+ liquid) region of the phase diagram.
(Although the maximum is observed very
near X
Mg= 0.5, there is no thermodynamic
reason for the maximum to occur exactly at
this composition.)
Another intermetallic phase, the ephase,
is also observed in the Ag–Mg system,
Fig. 1-15. The phase is associated with a
peritecticinvariant ABC at 744 K. The
Gibbs energy curves are shown schemati-
cally at the peritectic temperature in Fig.
1-15(c). One common tangent line can be
drawn to g
l
, g
b¢
and g
e
.
Suppose that a liquid alloy of composi-
tion X
Mg= 0.7 is cooled very slowly from
the liquid state. At a temperature just above
744 K a liquid phase of composition C and
26 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
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38. 1.5 Binary Phase Diagrams 27
Figure 1-14.CaO–SiO
2phase diagram at P= 1 bar (after Wu, 1990) and Gibbs energy curves at 1500 °C illus-
trating Gibbs energies of fusion and formation of the stoichiometric compound CaSiO
3.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

39. a b¢phase of composition A are observed
at equilibrium. At a temperature just below
744 K the two phases at equilibrium are b¢
of composition A and eof composition B.
The following invariant binary peritectic
reactionthus occurs upon cooling:
Liquid + b¢(solid) Æe(solid) (1-78)
This reaction occurs isothermally at 744 K
with all three phases at fixed compositions
(at points A, B and C). For an alloy with
overall composition between points A and
B the reaction proceeds until all the liquid
has been consumed. In the case of an alloy
with overall composition between B and C,
the b¢phase will be the first to be com-
pletely consumed.
Peritectic reactions occur upon cooling
with formation of the product solid (ein
this example) on the surface of the reactant
solid (b¢ ), thereby forming a coating which
can prevent further contact between the re-
actant solid and liquid. Further reaction
may thus be greatly retarded so that equi-
librium conditions can only be achieved by
extremely slow cooling.
The Gibbs energy curve for the e phase,
g
e
, in Fig. 1-15(c) rises more rapidly on ei-
ther side of its minimum than does the
Gibbs energy g
b¢
for the b¢ phase in Fig. 1-
15(a). As a result, the width of the single-
phase region over which the ephase exists
(sometimes called its range of stoichiome-
tryor homogeneity range) is narrower than
for the b¢phase.
In the upper panel of Fig. 1-14 for the
CaO–SiO
2system, Gibbs energy curves at
1500 °C for the liquid and CaSiO
3phases
are shown schematically. g
0.5 (CaSiO
3)rises
extremely rapidly on either side of its min-
imum. (We write g
0.5 (CaSiO
3)for 0.5 moles
of the compound in order to normalize to a
basis of one mole of components CaO and
SiO
2.) As a result, the points of tangency
Q
1and Q
2of the common tangents P
1Q
1
and P
2Q
2nearly (but not exactly) coincide.
Hence, the range of stoichiometry of the
CaSiO
3phase is very narrow (but never
zero). The two-phase regions labelled
(CaSiO
3+ liquid) in Fig. 1-14 are the two
sides of a two-phase region that passes
through a maximum at 1540 °C just as the
(b¢+ liquid) region passes through a maxi-
28 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-15.Ag–Mg phase diagram at P= 1 bar (af-
ter Hultgren at al., 1973) and Gibbs energy curves at
three temperatures.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

40. 1.5 Binary Phase Diagrams 29
mum in Fig. 1-15(d). Because the CaSiO
3
single-phase region is so narrow, we refer
to CaSiO
3as a stoichiometric compound.
Any deviation in composition from the
stoichiometric 1:1 ratio of CaO to SiO
2
results in a very large increase in Gibbs
energy.
The ephase in Fig. 1-15 is based on the
stiochiometry AgMg
3. The Gibbs energy
curve, Fig. 1-15(c), rises extremely rapidly
on the Ag side of the minimum, but some-
what less steeply on the Mg side. As a re-
sult, Ag is virtually insoluble in AgMg
3,
while Mg is sparingly soluble. Such a
phase with a narrow range of homogeneity
is often called a non-stoichiometric com-
pound. At low temperatures the b¢phase
exhibits a relatively narrow range of stoi-
chiometry about the 1:1 AgMg composi-
tion and can properly be called a com-
pound. However, at higher temperatures it
is debatable whether a phase with such a
wide range of composition should be called
a “compound”.
From Fig. 1-14 it can be seen that if stoi-
chiometric CaSiO
3is heated it will melt
isothermally at 1540 °C to form a liquid of
the same composition. Such a compound is
called congruently meltingor simply a con-
gruent compound. The compound Ca
2SiO
4
in Fig. 1-14 is congruently melting. The b¢
phase in Fig. 1-15 is also congruently melt-
ing at the composition of the liquidus/sol-
idus maximum.
It should be noted with regard to the con-
gruent melting of CaSiO
3in Fig. 1-14 that
the limiting slopes dT/dXof both branches
of the liquidus at the congruent melting
point (1540 °C) are zero since we are really
dealing with a maximum in a two-phase re-
gion.
The AgMg
3(e) compound in Fig. 1-15 is
said to melt incongruently. If solid AgMg
3
is heated it will melt isothermally at 744 K
by the reverse of the peritectic reaction,
Eq. (1-78), to form a liquid of composition
C and another solid phase, b¢, of composi-
tion A.
Another example of an incongruent
compound is Ca
3Si
2O
7in Fig. 1-14, which
melts incongruently (or peritectically) to
form liquid and Ca
2SiO
4at the peritectic
temperature of 1469 °C.
An incongruent compound is always as-
sociated with a peritectic. However, the
converse is not necessarily true. A peritec-
tic is not always associated with an inter-
mediate phase. See, for example, Fig. 1-
13(i).
For purposes of phase diagram calcula-
tions involving stoichiometric compounds
such as CaSiO
3, we may, to a good approx-
imation, consider the Gibbs energy curve,
g
0.5 (CaSiO
3), to have zero width. All that
is then required is the value of g
0.5 (CaSiO
3)
at the minimum. This value is usually
expressed in terms of the Gibbs energy
of fusion of the compound, Dg
0
f (0.5 CaSiO
3)
or the Gibbs energy of formation
Dg
0
form (0.5 CaSiO
3)of the compound from
the pure solid components CaO and SiO
2
according to the reaction: 0.5 CaO(sol) +
0.5 SiO
2(sol) = 0.5 CaSiO
3(sol). Both these
quantities are interpreted graphically in
Fig. 1-14.
1.5.11 Limited Mutual Solubility –
Ideal Henrian Solutions
In Sec. 1.5.6, the region of two solids in
the MgO–CaO phase diagram of Fig. 1-12
was described as a miscibility gap. That is,
only one continuous g
s
curve was assumed.
If, somehow, the appearance of the liquid
phase could be suppressed, then the two
solvus lines in Fig. 1-12, when projected
upwards, would meet at a critical point
above which one continuous solid solution
would exist at all compositions.www.iran-mavad.com
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41. Such a description is justifiable only if
the pure solid components have the same
crystal structure, as is the case for MgO
and CaO. However, consider the Ag–Mg
system, Fig. 1-15, in which the terminal
(Ag) solid solution is face-centered-cubic
and the terminal (Mg) solid solution is hex-
agonal-close-packed. In this case, one con-
tinuous curve for g
s
cannot be drawn. Each
solid phase must have its own separate
Gibbs energy curve, as shown schemati-
cally in Fig. 1-15(b) for the h.c.p. (Mg)
phase at 800 K. In this figure, g
Mg
0 (h.c.p.)and
g
Ag
0 (f.c.c.)are the standard molar Gibbs ener-
gies of pure h.c.p. Mg and pure f.c.c. Ag,
while g
Ag
0 (h.c.p.-Mg)is the standard molar
Gibbs energy of pure (hypothetical) h.c.p.
Ag in the h.c.p. (Mg) phase.
Since the solubility of Ag in the h.c.p.
(Mg) phase is limited we can, to a good ap-
proximation, describe it as a Henrian ideal
solution. That is, when a solution is suffi-
ciently dilute in one component, we can ap-
proximate g
E
solute
=RTlng
soluteby its value
in an infinitely dilute solution. That is, if
X
soluteis small we set g
solute=g
0
solute
where
g
0
solute
is the Henrian activity coefficientat
X
solute= 0. Thus, for sufficiently dilute solu-
tions we assume that
g
soluteis independent
of composition. Physically, this means that
in a very dilute solution there is negligible
interaction among solute particles because
they are so far apart. Hence, each addi-
tional solute particle added to the solution
produces the same contribution to the ex-
cess Gibbs energy of the solution and so g
E
-
solute=dG
E
/dn
solute= constant.
From the Gibbs–Duhem equation, Eq.
(1-56), if dg
E
solute
= 0, then dg
E
solvent
=0.
Hence, in a Henrian solution
g
soluteis also
constant and equal to its value in an infi-
nitely dilute solution. That is,
g
solute=1 and
the solvent behaves ideally. In summary
then, for dilute solutions (X
solvent≈1)
Henry’s Lawapplies:
g
solvent≈1
g
solute≈g
0
solute
= constant (1-79)
(Care must be exercised for solutions other
than simple substitutional solutions. Henry’s
Law applies only if the ideal activity is defined
correctly, as will be discussed in Sec. 1.10).
Treating, then, the h.c.p. (Mg) phase
in the Ag–Mg system (Fig. 1-15(b)) as a
Henrian solution we write:
g
h.c.p.
= (X
Agg
Ag
0 (f.c.c.)+ X
Mgg
Mg
0 (h.c.p.))
+ RT(X
Aglna
Ag+ X
Mglna
Mg)
= (X
Agg
Ag
0 (f.c.c.)+ X
Mgg
Mg
0 (h.c.p.)) (1-80)
+ RT(X
Agln (g
0
Ag
X
Ag) + X
MglnX
Mg)
where a
Agand g
0
Ag
are the activity and ac-
tivity coefficient of silver with respect to
pure f.c.c. silver as standard state. Let us
now combine terms as follows:
g
h.c.p.
= [X
Ag(g
Ag
0 (f.c.c.)+ RTlng
0
Ag
)
+ X
Mgg
Mg
0 (h.c.p.)] (1-81)
+ RT(X
AglnX
Ag+ X
MglnX
Mg)
Since
g
0
Ag
is independent of composition,
let us define:
g
Ag
0 (h.c.p.-Mg)= (g
Ag
0 (f.c.c.)+ RTlng
0
Ag
) (1-82)
From Eqs. (1-81) and (1-82) it can be seen
that, relative to g
Mg
0 (h.c.p.)and to the hypothet-
ical standard state g
Ag
0 (h.c.p.-Mg)defined in
this way, the h.c.p. solution is ideal. Eqs.
(1-81) and (1-82) are illustrated in Fig. 1-
15(b). It can be seen that as
g
0
Ag
becomes
larger, the point of tangency Nmoves
to higher Mg concentrations. That is, as
(g
Ag
0 (h.c.p.-Mg)–g
Ag
0 (f.c.c.)) becomes more posi-
tive, the solubility of Ag in h.c.p. (Mg) de-
creases.
It must be stressed that g
Ag
0 (h.c.p.-Mg)as de-
fined by Eq. (1-82) is solvent-dependent.
That is, g
Ag
0 (h.c.p.-Mg)is not the same as, say,
g
Ag
0 (h.c.p.-Cd)for Ag in dilute h.c.p. (Cd) solid
solutions.
30 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

42. 1.5 Binary Phase Diagrams 31
Henrian activity coefficients can usually
be expressed as functions of temperature:
RTln
g
i
0= a– bT (1-83)
where aand bare constants. If data are lim-
ited, it can further be assumed that b≈0 so
that RT ln
g
i
0≈constant.
1.5.12 Geometry of Binary Phase
Diagrams
The geometry of all types of phase dia-
grams of any number of components is
governed by the Gibbs Phase Rule.
Consider a system with Ccomponents in
which Pphases are in equilibrium. The
system is described by the temperature, the
total pressure and the composition of each
phase. In a C-component system, (C–1) in-
dependent mole fractions are required to
describe the composition of each phase
(because
SX
i=1). Hence, the total number
of variables required to describe the system
is [P(C–1) + 2]. However, as shown in Sec.
1.4.2, the chemical potential of any compo-
nent is the same in all phases (a, b, g,…)
since the phases are in equilibrium. That is:
g
i
a(T, P, X
1
a, X
2
a, X
3
a, …)
= g
i
b(T, P, X
1
b, X
2
b, X
3
b, …)
= g
i
g(T, P, X
1
g, X
2
g, X
3
g, …) = … (1-84)
where g
i
a(T, P, X
1
a, X
2
a, X
3
a, …) is a func-
tion of temperature, of total pressure, and
of the mole fractions X
1
a, X
2
a, X
3
a, … in
the aphase; and similarly for the other
phases. Thus there are C(P–1) indepen-
dent equations in Eq. (1-84) relating the
variables.
Let Fbe the differences between the
number of variables and the number of
equations relating them:
F= P(C– 1) + 2 – C(P–1)
F= C– P+ 2 (1-85)
This is the Gibbs Phase Rule. Fis called
the number of degrees of freedomor vari-
anceof the system and is the number of pa-
rameters which can and must be specified
in order to completely specify the state of
the system.
Binary temperature–composition phase
diagrams are plotted at a fixed pressure,
usually 1 bar. This then eliminates one de-
gree of freedom. In a binary system, C=2.
Hence, for binary isobaric T–Xdiagrams
the phase rule reduces to:
F= 3 – P (1-86)
Binary T–Xdiagrams contain single-
phase areas and two-phase areas. In the sin-
gle-phase areas, F= 3 – 1 = 2. That is, tem-
perature and composition can be specified
independently. These regions are thus
called bivariant. In two-phase regions,
F= 3 – 2 = 1. If, say, T is specified, then the
compositions of both phases are deter-
mined by the ends of the tie-lines. Two-
phase regions are thus termed univariant.
Note that the overall composition can be
varied within a two-phase region at con-
stant T, but the overall composition is not a
parameter in the sense of the phase rule.
Rather, it is the compositions of the indi-
vidual phases at equilibrium that are the
parameters to be considered in counting the
number of degrees of freedom.
When three phases are at equilibrium in
a binary system at constant pressure,
F= 3 – 3 = 0. Hence, the compositions of
all three phases, as well as T, are fixed.
There are two general types of three-phase
invariantsin binary phase diagrams. These
are the eutectic-typeand peritectic-type
invariants as illustrated in Fig. 1-16. Let
the three phases concerned be called a, b
and g, with bas the central phase as shown
in Fig. 1-16. The phases a, band gcan be
solid, liquid or gaseous. At the eutectic-
type invariant, the following invariant re
-www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

43. action occurs isothermally as the system is
cooled:
b Æ a+ g (1-87)
whereas at the peritectic-type invariant the
invariant reaction upon cooling is:
a+ g Æ b (1-88)
Some examples of eutectic-type invari-
ants are: (i) eutectics(Fig. 1-12) in which
a= solid
1, b= liquid, g= solid
2; the eutectic
reaction is lÆs
1+s
2; (ii)monotectics(Fig.
1-14) in which a= liquid
1, b= liquid
2, g=-
solid; the monotectic reaction is l
2Æl
1+s;
(iii) eutectoidsin which a =
solid
1, b= solid
2, g= solid
3; the eutectoid
reaction is s
2Æs
1+s
3; (iv) catatecticsin
which a= liquid, b= solid
1, g= solid
2; the
catatectic reaction is s
1Æl+s
2.
Some examples of peritectic-type invari-
ants are: (i) peritectics(Fig. 1-15) in which
a= liquid, b= solid
1, g= solid
2. The peri-
tectic reaction is l + s
2Æs
1; (ii) syntectics
(Fig. 1-13(k)) in which a= liquid
1, b=
solid, g= liquid
2. The syntectic reaction is
l
1+l
2Æs; (iii) peritectoids in which a=
solid
1, b= solid
2, g= solid
3. The peritec-
toid reaction is s
1+s
3Æs
2.
An important rule of construction which
applies to invariants in binary phase dia-
grams is illustrated in Fig. 1-16. This ex-
tension rulestates that at an invariant the
extension of a boundary of a two-phase re-
gion must pass into the adjacent two-phase
region and not into a single-phase region.
Examples of both correct and incorrect
constructions are given in Fig. 1-16. To
understand why the “incorrect extensions”
shown are not right consider that the (a+g)
phase boundary line indicates the composi-
tion of the g-phase in equilibrium with the
a-phase, as determined by the common
tangent to the Gibbs energy curves. Since
there is no reason for the Gibbs energy
curves or their derivatives to change dis-
continuously at the invariant temperature,
the extension of the (a+g) phase boundary
also represents the stable phase boundary
under equilibrium conditions. Hence, for
this line to extend into a region labeled as
single-phase gis incorrect.
Two-phase regions in binary phase dia-
grams can terminate: (i) on the pure com-
ponent axes (at X
A= 1 or X
B= 1) at a trans-
formation point of pure A or B; (ii) at a
critical point of a miscibility gap; (iii) at an
invariant. Two-phase regions can also ex-
hibit maxima or minima. In this case, both
phase boundaries must pass through their
maximum or minimum at the same point as
shown in Fig. 1-16.
All the geometrical unitsof construction
of binary phase diagrams have now been
discussed. The phase diagram of a binary
alloy system will usually exhibit several of
these units. As an example, the Fe–Mo
phase diagram (Kubaschewski, 1982) is
32 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-16.Some geometrical units of binary phase
diagrams, illustrating rules of construction.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

44. 1.5 Binary Phase Diagrams 33
shown in Fig. 1-17. The invariants in this
system are peritectics at 1540, 1488 and
1450 °C; eutectoids at 1235 and 1200 °C;
peritectoids at 1370 and 950 °C. The two-
phase (liquid +g) region passes through a
minimum at X
Mo= 0.2.
Between 910 °C and 1390 °C is a two-
phase (a +g) g-loop. Pure Fe adopts the
f.c.c. gstructure between these two temper-
atures but exists as the b.c.c. aphase at
higher and lower temperatures. Mo, how-
ever, is more soluble in the b.c.c. than
in the f.c.c. structure. That is, g
Mo
0 (b.c.c.-Fe)
<g
Mo
0 (f.c.c.-Fe)as discussed in Sec. 1.5.11.
Therefore, small additions of Mo stabilize
the b.c.c. structure.
In the CaO–SiO
2phase diagram, Fig.
1-14, we observe eutectics at 1439, 1466
and 2051°C; a monotectic at 1692 °C; and
a peritectic at 1469 °C. The compound
Ca
3SiO
5dissociates upon heating to CaO
and Ca
2SiO
4by a peritectoid reaction at
1789 °C and dissociates upon cooling to
CaO and Ca
2SiO
4by a eutectoid reaction at
1250 °C. Maxima are observed at 2130 and
1540 °C. At 1470 °C there is an invariant
associated with the tridymiteÆcristobalite
transition of SiO
2. This is either a peritec-
tic or a catatectic depending upon the rela-
tive solubility of CaO in tridymite and cris-
tobalite. However, these solubilities are
very small and unknown.
Figure 1-17.Fe–Mo
phase diagram at P= 1 bar
(Kubaschewski, 1982).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

45. 1.6 Application of
Thermodynamics to Phase
Diagram Analysis
1.6.1 Thermodynamic/Phase Diagram
Optimization
In recent years the development of
solution models, numerical methods and
computer software has permitted a quanti-
tative application of thermodynamics to
phase diagram analysis. For a great many
systems it is now possible to perform a
simultaneous critical evaluation of avail-
able phase diagram measurements and of
available thermodynamic data (calorimet-
ric data, measurements of activities, etc.)
with a view to obtaining optimized equa-
tions for the Gibbs energies of each phase
which best represent all the data. These
equations are consistent with thermody-
namic principles and with theories of solu-
tion behavior.
The phase diagram can be calculated
from these thermodynamic equations, and
so one set of self-consistent equations de-
scribes all the thermodynamic properties
and the phase diagram. This technique of
analysis greatly reduces the amount of ex-
perimental data needed to fully character-
ize a system. All data can be tested for
internal consistency. The data can be inter-
polated and extrapolated more accurately
and metastable phase boundaries can be
calculated. All the thermodynamic proper-
ties and the phase diagram can be repre-
sented and stored by means of a small set
of coefficients.
Finally, and most importantly, it is often
possible to estimate the thermodynamic
properties and phase diagrams of ternary
and higher-order systems from the assessed
parameters for their binary sub-systems, as
will be discussed in Sec. 1.11. The analysis
of binary systems is thus the first and most
important step in the development of data-
bases for multicomponent systems.
1.6.2 Polynomial Representation
of Excess Properties
Empirical equations are required to ex-
press the excess thermodynamic properties
of the solution phases as functions of com-
position and temperature. For many simple
binary substitutional solutions, a good rep-
resentation is obtained by expanding the
excess enthalpy and entropy as polynomi-
als in the mole fractions X
Aand X
Bof the
components:
h
E
= X
AX
B[h
0+ h
1(X
B– X
A) (1-89)
+ h
2(X
B– X
A)
2
+ h
3(X
B– X
A)
3
+ …]
s
E
= X
AX
B[s
0+ s
1(X
B– X
A) (1-90)
+ s
2(X
B– X
A)
2
+ s
3(X
B– X
A)
3
+ …]
where the h
iand s
iare empirical coeffi-
cients. As many coefficients are used as
are required to represent the data in a
given system. For most systems it is a good
approximation to assume that the coeffi-
cients h
iand s
iare independent of tempera-
ture.
If the series are truncated after the first
term, then:
g
E
= h
E
– Ts
E
= X
AX
B(h
0– Ts
0) (1-91)
This is the equation for a regular solution
discussed in Sec. 1.5.7. Hence, the polyno-
mial representation can be considered to be
an extension of regular solution theory.
When the expansions are written in terms
of the composition variable (X
B–X
A), as in
Eqs. (1-89) and (1-90), they are said to be
in Redlich–Kister form. Other equivalent
polynomial expansions such as orthogonal
Legendre series have been discussed by
Pelton and Bale (1986).
Differentiation of Eqs. (1-89) and (1-90)
and substitution into Eq. (1-55) yields the
34 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
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46. 1.6 Application of Thermodynamics to Phase Diagram Analysis 35
following expansions for the partial excess
enthalpies and entropies:
h
A
E= X
B
2S
i=0
h
i[(X
B– X
A)
i
– 2iX
A(X
B–X
A)
i–1
] (1-92)
h
B
E= X
A
2S
i=0
h
i[(X
B– X
A)
i
+ 2iX
B(X
B–X
A)
i–1
] (1-93)
s
A
E= X
B
2S
i=0
s
i[(X
B– X
A)
i
– 2iX
A(X
B–X
A)
i–1
] (1-94)
s
B
E= X
A
2S
i=0
s
i[(X
B– X
A)
i
+ 2iX
B(X
B–X
A)
i–1
] (1-95)
Partial excess Gibbs energies, g
i
E, are
then given by Eq. (1-52).
Eqs. (1-89) and (1-90), being based upon
regular solution theory, give an adequate
representation for most simple substitu-
tional solutions in which deviations from
ideal behavior are not too great. In other
cases, more sophisticated models are re-
quired, as discussed in Sec. 1.10.
1.6.3 Least-Squares Optimization
Eqs. (1-89), (1-90) and (1-92) to (1-95)
are linear in terms of the coefficients.
Through the use of these equations, all
integral and partial excess properties (g
E
,
h
E
, s
E
, g
i
E, h
i
E, s
i
E) can be expressed by
linear equations in terms of the one set of
coefficients {h
i, s
i}. It is thus possible to
include all available experimental data for
a binary phase in one simultaneous linear
least-squares optimization. Details have
been discussed by Bale and Pelton (1983),
Lukas et al. (1977) and Dörner et al.
(1980).
The technique of coupled thermody-
namic/phase diagram analysis is best illus-
trated by examples.
The phase diagram of the LiF–NaF
system is shown in Fig. 1-18. Data points
measured by Holm (1965) are shown on
the diagram. The Gibbs energy of fusion of
each pure component at temperature Tis
given by:
where Dh
0
f(
T
f)is the enthalpy of fusion at
the melting point T
f, and c
l
p
and c
s
p
are the
heat capacities of the pure liquid and solid.
The following values are taken from Barin
et al. (1977):
Dg
0
f (LiF)
= 14.518 + 128.435T
+ 8.709 ¥10
–3
T
2
– 21.494TlnT
– 2.65 ¥10
5
T
–1
J/mol (1-97)
Dg
0
f (NaF)
= 10.847 + 156.584T
+ 4.950 ¥10
–3
T
2
– 23.978TlnT
– 1.07 ¥10
5
T
–1
J/mol (1-98)
Thermodynamic properties along the liq-
uidus and solidus are related by equations
like Eqs. (1-64) and (1-65). Taking the
ideal activities to be equal to the mole frac-
tions:
RTlnX
i
l– RTlnX
i
s+ g
i
E (l)– g
i
E(s)
= – Dg
0
f(i)
(1-99)
DDg
f
0
f f
p
l
p
s=
d(1-)
f
f
hTT
cc TT
T
T
T
()(/)
()(/)
0
1
11 96
−
+−−
∫
Figure 1-18.LiF–NaF phase diagram at P= 1 bar
calculated from optimized thermodynamic parame-
ters (Sangster and Pelton, 1987). Points are experi-
mental from Holm (1965). Dashed line is theoretical
limiting liquidus slope for negligible solid solubility.www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

47. where i= LiF or NaF. Along the LiF-rich
liquidus, the liquid is in equilibrium with
essentially pure solid LiF. Hence, X
s
LiF
=1
and g
Lif
E(s)= 0. Eq. (1-99) then reduces to:
RTlnX
l
LiF
+ g
Lif
E (l)= – Dg
0
f (LiF)
(1-100)
From experimental values of X
l
LiF
on the
liquidus and with Eq. (1-97) for Dg
0
f(LiF)
,
values of g
Lif
E (l)at the measured liquidus
points can be calculated from Eq. (1-100).
Along the NaF-rich solidus the solid so-
lution is sufficiently concentrated in NaF
that Henrian behavior (Sec. 1.5.11) can be
assumed. That is, for the solvent, g
NaF
E(s)=0.
Hence, Eq. (1-99) becomes:
RTlnX
(l)
NaF
– RTlnX
(s)
NaF
+ g
NaF
E(l)
= – Dg
0
f (NaF)
(1-101)
Thus, from the experimental liquidus and
solidus compositions and with the Gibbs
energy of fusion from Eq. (1-98), values of
g
NaF
E(l)can be calculated at the measured liq-
uidus points from Eq. (1-101).
Finally, enthalpies of mixing, h
E
, in the
liquid have been measured by calorimetry
by Hong and Kleppa (1976).
Combining all these data in a least-
squares optimization, the following expres-
sions for the liquid were obtained by Sang-
ster and Pelton (1987):
h
E (l)
= X
LiFX
NaF (1-102)
¥[– 7381 + 184 (X
NaF– X
LiF)] J/mol
s
E(l)
= X
LiFX
NaF (1-103)
¥[– 2.169 – 0.562 (X
NaF– X
LiF)] J/mol
Eqs. (1-102) and (1-103) then permit all
other integral and partial properties of the
liquid to be calculated.
For the NaF-rich Henrian solid solution,
the solubility of LiF has been measured by
Holm (1965) at the eutectic temperature
where the NaF-rich solid solution is in
equilibrium with pure solid LiF. That is,
a
LiF=1 with respect to pure solid LiF as
standard state. In the Henrian solution at
saturation,
a
LiF= g
0
LiF
X
LiF= g
0
LiF
(1 – 0.915) = 1
Hence, the Henrian activity coefficient
in the NaF-rich solid solution at 649 °C
is
g
0
LiF
= 11.76. Since no solubilities have
been measured at other temperatures, we
assume that:
RTln
g
0
LiF
= R(922) ln (11.76) (1-104)
= 18 900 J/mol = constant
Using the notation of Eq. (1-82):
g
LiF
0 (s, NaF)= g
LiF
0(s)+ 18 900 J/mol (1-105)
where g
LiF
0(s)is the standard Gibbs energy of
solid LiF, and g
LiF
0 (s, NaF)is the hypothetical
standard Gibbs energy of LiF dissolved in
solid NaF.
The phase diagram drawn in Fig. 1-18
was calculated from Eqs. (1-97) to (1-104).
Complete details of the analysis of the
LiF–NaF system are given by Sangster and
Pelton (1987).
As a second example of thermodynamic/
phase diagram optimization, consider the
Cd–Na system. The phase diagram, with
points measured by several authors (Math-
ewson, 1906; Kurnakow and Kusnetzow,
1907; Weeks and Davies, 1964) is shown in
Fig. 1-19.
From electromotive force measurements
on alloy concentration cells, several au-
thors have measured the activity coeffi-
cient of Na in liquid alloys. The data
are shown in Fig. 1-20 at 400 °C. From
the temperature dependence of g
E
Na
=
RTln
g
Na, the partial enthalpy of Na in the
liquid was obtained via Eq. (1-52). The re-
sults are shown in Fig. 1-21. Also, h
E
of the
liquid has been measured by Kleinstuber
(1961) by direct calorimetry. These ther-
modynamic data for g
E
Na
, h
E
Na
and h
E
were
36 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

48. 1.6 Application of Thermodynamics to Phase Diagram Analysis 37
Figure 1-19.Cd–Na phase diagram at P= 1 bar calculated from optimized thermodynamic parameters (Re-
printed from Pelton, 1988a). ∫ Kurnakow and Kusnetzow (1907), ⎛Mathewson (1906), ×Weeks and Davies
(1964).
Figure 1-20.Sodium ac-
tivity coefficient in liquid
Cd–Na alloys at 400°C.
Line is calculated from
optimized thermodynamic
parameters (Reprinted
from Pelton, 1988a).
⎝Hauffe (1940),
⎜Lantratov and
Mikhailova (1971),
⎛Maiorova et al. (1976),
⎞Alabyshev and
Morachevskii (1957),
∫Bartlett et al. (1970).www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

49. optimized simultaneously (Pelton, 1988a)
to obtain the following expressions for h
E
and s
E
of the liquid:
h
E (l)
= X
CdX
Na[– 12 508 + 20 316 (1-106)
¥(X
Na– X
Cd) – 8714 (X
Na– X
Cd)
2
] J/mol
s
E (l)
= X
CdX
Na[– 15.452 + 15.186 (1-107)
¥(X
Na– X
Cd) – 10.062 (X
Na– X
Cd)
2
– 1.122 (X
Na– X
Cd)
3
] J/mol K
Eq. (1-106) reproduces the calorimetric
data within 200 J/mol
–1
. Eqs. (1-52), (1-
58), (1-93) and (1-95) can be used to calcu-
late h
E
Na
and g
Na. The calculated curves are
compared to the measured points in Figs.
1-20 and 1-21.
For the two compounds, Gibbs energies
of fusion were calculated (Pelton, 1988a)
so as to best reproduce the measured phase
diagram:
Dg
0
f (1/13 Cd
11Na
2)= 6816 – 10.724TJ/g-atom
(1-108)
Dg
0
f(1/3Cd
2Na)= 8368 – 12.737TJ/g-atom
(1-109)
The optimized enthalpies of fusion of 6816
and 8368 J/g-atom agree within error lim-
its with the values of 6987 and 7878 J/g-
atom measured by Roos (1916). (See Fig.
1-14 for an illustration of the relation
between the Gibbs energy of fusion of a
compound and the phase diagram.)
The phase diagram shown in Fig. 1-19
was calculated from Eqs. (1-106) to (1-
109) along with the Gibbs energies of fu-
sion of Cd and Na taken from the literature
(Chase, 1983). Complete details of the
analysis of the Cd–Na system are given by
Pelton (1988a).
It can thus be seen that one simple set of
equations can simultaneously and self-con-
sistently describe all the thermodynamic
properties and the phase diagram of a bi-
nary system.
The exact optimization procedure will
vary from system to system depending
upon the type and accuracy of the avail-
able data, the number of phases present, the
extent of solid solubility, etc. A large num-
ber of optimizations have been published
in the Calphad Journal(Pergamon) since
1977.
38 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-21.Partial excess enthalpy of sodium in
liquid Cd–Na alloys. Line is calculated from opti-
mized thermodynamic parameters (Reprinted from
Pelton, 1988a). Lantratov and Mikhailova (1971),
Maiorova et al. (1976), Bartlett et al. (1970).www.iran-mavad.com
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50. 1.7 Ternary and Multicomponent Phase Diagrams 39
1.6.4 Calculation of Metastable Phase
Boundaries
In the Cd–Na system just discussed, the
liquid exhibits positive deviations from
ideal mixing. That is, g
E(l)
> 0. This fact is
reflected in the very flat liquidus in Fig.
1-19 as was discussed in Sec. 1.5.9.
By simply not including any solid phases
in the calculation, the metastable liquid
miscibility gap as well as the spinodal
curve (Sec. 1.5.5) can be calculated as
shown in Fig. 1-19. These curves are im-
portant in the formation of metallic glasses
by rapid quenching.
Other metastable phase boundaries such
as the extension of a liquidus curve below a
eutectic can also be calculated thermody-
namically by simply excluding one or more
phases during the computations.
1.7 Ternary and Multicomponent
Phase Diagrams
This section provides an introduction to
ternary phase diagrams. For a more de-
tailed treatment, see Prince (1966); Ricci
(1964); Findlay (1951); or West (1965).
1.7.1 The Ternary Composition Triangle
In a ternary system with components
A–B–C, the sum of the mole fractions is
unity, (X
A+X
B+X
C) = 1. Hence, there are
two independent composition variables. A
representation of composition, symmetri-
cal with respect to all three components,
may be obtained with the equilateral “com-
position triangle” as shown in Fig. 1-22 for
the Bi–Sn–Cd system. Compositions at
the corners of the triangle correspond to the
pure components. Along the edges of the
triangle compositions corresponding to the
three binary subsystems Bi–Sn, Sn–Cd
and Cd–Bi are found. Lines of constant
mole fraction X
Biare parallel to the Sn–Cd
edge, while lines of constant X
Snand X
Cd
are parallel to the Cd–Bi and Bi–Sn edges
respectively. For example, at point ain Fig.
1-22, X
Bi= 0.05, X
Sn= 0.45 and X
Cd= 0.50.
Similar equilateral composition triangles
can be drawn with coordinates in terms of
wt.% of the three components.
1.7.2 Ternary Space Model
A ternary temperature–composition
“phase diagram” at constant total pressure
may be plotted as a three-dimensional
“space model” within a right triangular
prism with the equilateral composition tri-
angle as base and temperature as vertical
axis. Such a space model for a simple eu-
tectic ternary system A–B–C is illustrated
in Fig. 1-23. On the three vertical faces of
the prism we find the phase diagrams of the
three binary subsystems, A–B, B–C and
C–A which, in this example, are all simple
eutectic binary systems. The binary eutec-
tic points are e
1, e
2and e
3. Within the
prism we see three liquidus surfacesde-
scending from the melting points of pure
A, B and C. Compositions on these sur-
faces correspond to compositions of liquid
in equilibrium with A-, B- and C-rich solid
phases.
In a ternary system at constant pressure,
the Gibbs phase rule, Eq. (1-85), becomes:
F= 4 – P (1-110)
When the liquid and one solid phase are in
equilibrium P= 2. Hence F= 2 and the
system is bivariant. A ternary liquidus is
thus a two-dimensional surface. We may
choose two variables, say Tand one
composition coordinate of the liquid, but
then the other liquid composition coordi-
nate and the composition of the solid are
fixed.www.iran-mavad.com
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51. 40 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-22.Projection of the liquidus surface of the Bi–Sn–Cd system onto the ternary composition triangle
(after Bray et al., 1961–1962). Small arrows show the crystallization path of an alloy of overall composition at
point a. (Reprinted from Pelton, 1996.)
Figure 1-23.Perspective view of ternary space
model of a simple eutectic ternary system. e
1, e
2, e
3
are the binary eutectics and Eis the ternary eutectic.
The base of the prism is the equilateral composition triangle. (Reprinted from Pelton, 1983.)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

52. 1.7 Ternary and Multicomponent Phase Diagrams 41
The A- and B-liquidus surfaces in Fig. 1-
23 intersect along the line e
1E. Liquids
with compositions along this line are there-
fore in equilibrium with A-rich and B-rich
solid phases simultaneously. That is, P=3
and so F–1. Such “valleys” are thus called
univariant lines. The three univariant lines
meet at the ternary eutectic point Eat
which P= 4 and F= 0. This is an invariant
point since the temperature and the compo-
sitions of all four phases in equilibrium are
fixed.
1.7.3 Polythermal Projections
of Liquidus Surfaces
A two-dimensional representation of the
ternary liquidus surface may be obtained as
an orthogonal projection upon the base
composition triangle. Such a polythermal
projection of the liquidus of the Bi–Sn–Cd
system (Bray et al., 1961–62) is shown in
Fig. 1-22. This is a simple eutectic ternary
system with a space model like that shown
in Fig. 1-23. The constant temperature
lines on Fig. 1-22 are called liquidus iso-
therms. The univariant valleys are shown
as heavier lines. By convention, the large
arrows indicate the directions of decreas-
ing temperature along these lines.
Let us consider the sequence of events
occurring during the equilibrium cooling
from the liquid of an alloy of overall com-
position ain Fig. 1-22. Point alies within
the field of primary crystallizationof Cd.
That is, it lies within the composition re-
gion in Fig. 1-22 in which Cd-rich solid
will be the first solid to precipitate upon
cooling. As the liquid alloy is cooled, the
Cd-liquidus surface is reached at T≈465 K
(slightly below the 473 K isotherm). A
solid Cd-rich phase begins to precipitate at
this temperature. Now, in this particular
system, Bi and Sn are nearly insoluble in
solid Cd, so that the solid phase is virtually
pure Cd. (Note that this fact cannot be de-
duced from Fig. 1-22 alone.) Therefore, as
solidification proceeds, the liquid becomes
depleted in Cd, but the ratio X
Sn/X
Biin the
liquid remains constant. Hence, the compo-
sition path followed by the liquid (its crys-
tallization path) is a straight line passing
through point aand projecting to the Cd-
corner of the triangle. This crystallization
path is shown on Fig. 1-22 as the line ab.
In the general case in which a solid solu-
tion rather than a pure component or stoi-
chiometric compound is precipitating, the
crystallization path will not be a straight
line. However, for equilibrium cooling, a
straight line joining a point on the crystal-
lization path at any Tto the overall compo-
sition point awill extend through the com-
position, on the solidus surface, of the solid
phase in equilibrium with the liquid at that
temperature.
When the composition of the liquid has
reached point bin Fig. 1-22 at T≈435 K,
the relative proportions of the solid Cd and
liquid phases at equilibrium are given by
the lever ruleapplied to the tie-line dab:
(moles of liquid)/(moles of Cd) = da/ab.
Upon further cooling, the liquid composi-
tion follows the univariant valley from bto
Ewhile Cd and Sn-rich solids coprecipitate
as a binary eutectic mixture. When the
liquidus composition attains the ternary eu-
tectic composition Eat T≈380 K the invar-
iant ternary eutectic reaction occurs:
liquid Æs
1+ s
2+ s
3 (1-111)
where s
1, s
2and s
3are the three solid
phases and where the compositions of all
four phases (as well as T) remain fixed un-
til all liquid is solidified.
In order to illustrate several of the fea-
tures of polythermal projections of liquidus
surfaces, a projection of the liquidus of a
hypothetical system A–B–C is shown in
Fig. 1-24. For the sake of simplicity, iso-www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

53. therms are not shown, only the univariant
lines with arrows to show the directions of
decreasing temperature. The binary sub-
systems A–B and C–A are simple eutectic
systems, while the binary subsystem B–C
contains one congruent binary phase, e,
and one incongruent binary phase, d, as
shown in the insert in Fig. 1-24. The letters
eand pindicate binary eutectic and peritec-
tic points. The e and dphases are called bi-
nary compoundssince they have composi-
tions within a binary subsystem. Two ter-
nary compounds, hand z, with composi-
tions within the ternary triangle, as indi-
cated in Fig. 1-24, are also found in this
system. All compounds, as well as pure
solid A, B and C (the “a, band g” phases),
are assumed to be stoichiometric (i.e., there
is no solid solubility). The fields of pri-
mary crystallization of all the solids are in-
dicated in parentheses in Fig. 1-24. The
composition of the ephase lies within its
field, since eis a congruent compound,
while the composition of the dphase lies
outside of its field since dis incongruent.
Similarly for the ternary compounds, his a
congruently melting compound while zis
incongruent. For the congruent compound
h, the highest temperature on the hliquidus
occurs at the composition of h.
The univariant lines meet at a number of
ternary eutectics E(three arrows converg-
ing), a ternary peritectic P(one arrow en-
tering, two arrows leaving the point), and
several ternary quasi-peritectics P¢(two
arrows entering, one arrow leaving). Two
saddle points s are also shown. These are
points of maximum Talong the univariant
line but of minimum Ton the liquidus sur-
face along a section joining the composi-
tions of the two solids. For example, s
1is at
a maximum along the univariant E
1P¢
3, but
is a minimum point on the liquidus along
the straight line zs
1h.
42 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-24.Projection of
the liquidus surface of a
system A–B–C. The binary
subsystems A–B and C–A
are simple eutectic systems.
The binary phase diagram
B–C is shown in the insert.
All solid phases are assumed
pure stoichiometric compo-
nents or compounds. Small
arrows show crystallization
paths of alloys of composi-
tions at points aand b. (Re-
printed from Pelton, 1983.)www.iran-mavad.com
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54. 1.7 Ternary and Multicomponent Phase Diagrams 43
Let us consider the events occurring dur-
ing cooling from the liquid of an alloy of
overall composition ain Fig. 1-24. The pri-
mary crystallization product will be the e
phase. Since this is a pure stoichiometric
solid the crystallization path of the liquid
will be along a straight line passing
through aand extending to the composition
of eas shown in the figure.
Solidification of e continues until the
liquid attains a composition on the univari-
ant valley. Thereafter the liquid composi-
tion follows the valley towards the point P¢
1
in co-existence with eand z. At point P ¢
1
the invariant ternary quasi-peritectic reac-
tion occurs isothermally:
liquid + eÆd+ z (1-112)
Since there are two reactants in a quasi-
peritectic reaction, there are two possible
outcomes: (i) the liquid is completely con-
sumed before the ephase; in this case, so-
lidification will be complete at the point P¢
1;
(ii) eis completely consumed before the
liquid. In this case, solidification will con-
tinue with decreasing Talong the univari-
ant line P ¢
1E
1with co-precipitation of dand
zuntil, at E, the liquid will solidify eutecti-
cally (liquidÆd+ z+h). To determine
whether outcome (i) or (ii) occurs, we use
the mass balance criterion that, for three-
phase equilibrium, the overall composition
amust always lie within the tie-triangle
formed by the compositions of the three
phases. Now, the triangle joining the com-
positions of d, eand zdoes not contain the
point a, but the triangle joining the compo-
sitions of d, zand liquid at P ¢
1does contain
the point a. Hence, outcome (ii) occurs.
An alloy of overall composition bin Fig.
1-24 solidifies with eas primary crystal-
lization product until the liquid composi-
tion contacts the univariant line. There-
after, co-precipitation of eand boccurs
with the liquid composition following the
univariant valley until the liquid reaches
the peritectic composition P. The invariant
ternary peritectic reactionthen occurs iso-
thermally:
liquid + e+ bÆz (1-113)
Since there are three reactants, there are
three possible outcomes: (i) the liquid is
consumed before either eor band solidifi-
cation terminates at P; (ii) eis consumed
first, solidification then continues along
the path PP¢
3; or (iii) bis consumed first
and solidification continues along the path
PP¢
1. Which outcome occurs depends on
whether the overall composition blies
within the tie-triangle (i) ebz; (ii) bzP , or
(iii) ezP. In the example shown, outcome
(i) will occur.
1.7.4 Ternary Isothermal Sections
Isothermal projections of the liquidus
surface do not provide information on the
compositions of the solid phases at equilib-
rium. However, this information can be
presented at any one temperature on an iso-
thermal sectionsuch as that shown for the
Bi–Sn–Cd system at 423 K in Fig. 1-25.
This phase diagram is a constant tempera-
ture slice through the space model of Fig.
1-23.
The liquidus lines bordering the one-
phase liquid region of Fig. 1-25 are identi-
cal to the 423 K isotherms of the projection
in Fig. 1-22. Point cin Fig. 1-25 is point c
on the univariant line in Fig. 1-22. An alloy
with overall composition in the one-phase
liquid region of Fig. 1-25 at 423 K will
consist of a single liquid phase. If the over-
all composition lies within one of the two-
phase regions, then the compositions of the
two phases are given by the ends of the tie-
linewhich passes through the overall com-
position. For example, a sample with over-
all composition pin Fig. 1-25 will consistwww.iran-mavad.com
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55. of a liquid of composition qon the liquidus
and a solid Bi-rich alloy of composition r
on the solidus. The relative proportions of
the two phases are given by the lever rule:
(moles of liquid)/(moles of solid) =pr/pq,
where prand pqare the lengths of the tie-
line segments.
In the case of solid Cd, the solid phase is
nearly pure Cd, so all tie-lines of the (Cd+
liquid) region converge nearly to the corner
of the triangle. In the case of Bi- and Sn-
rich solids, some solid solubility is ob-
served. (The actual extent of this solubility
is somewhat exaggerated in Fig. 1-25 for
the sake of clarity of presentation.) Alloys
with overall compositions rich enough in
Bi or Sn to lie within the single-phase (Sn)
or (Bi) regions of Fig. 1-25 will consist at
423 K of single-phase solid solutions. Al-
loys with overall compositions at 423 K in
the two-phase (Cd + Sn) region will consist
of two solid phases.
Alloys with overall compositions within
the three-phase triangle dcfwill, at 423 K,
consist of three phases: solid Cd- and Sn-
rich solids with compositions at dand fand
44 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-25.Isothermal section of the Bi–Sn–Cd system at 423 K at P= 1 bar (after Bray et al., 1961–1962).
Extents of solid solubility in Bi and Sn have been exaggerated for clarity of presentation. (Reprinted from Pel-
ton, 1996.)www.iran-mavad.com
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56. 1.7 Ternary and Multicomponent Phase Diagrams 45
liquid of composition c. To understand this
better, consider an alloy of composition a
in Fig. 1-25, which is the same composi-
tion as the point a in Fig. 1-22. In Sec.
1.7.3 we saw that when an alloy of this
composition is cooled, the liquid follows
the path abin Fig. 1-22 with primary pre-
cipitation of Cd and then follows the uni-
variant line with co-precipitation of Cd and
Sn so that at 423 K the liquid is at the com-
position point c, and two solid phases are in
equilibrium with the liquid.
1.7.4.1 Topology of Ternary Isothermal
Sections
At constant temperature the Gibbs en-
ergy of each phase in a ternary system is
represented as a function of composition
by a surface plotted in a right triangular
prism with Gibbs energy as vertical axis
and the composition triangle as base. Just
as the compositions of phases at equilib-
rium in binary systems are determined
by the points of contact of a common tan-
gent line to their isothermal Gibbs energy
curves, so the compositions of phases at
equilibrium in a ternary system are given
by the points of contact of a common tan-
gent plane to their isothermal Gibbs energy
surfaces. A common tangent plane can
contact two Gibbs energy surfaces at an in-
finite number of pairs of points, thereby
generating an infinite number of tie-lines
within a two-phase region on an isothermal
section. A common tangent plane to three
Gibbs energy surfaces contacts each sur-
face at a unique point, thereby generating a
three-phase tie-triangle.
Hence, the principal topological units of
construction of an isothermal ternary phase
diagram are three-phase (a+b+g) tie-tri-
angles as in Fig. 1-26 with their accompa-
nying two-phase and single-phase areas.
Each corner of the tie-triangle contacts a
single-phase region, and from each edge of
the triangle there extends a two-phase re-
gion. The edge of the triangle is a limiting
tie-line of the two-phase region.
For overall compositions within the tie-
triangle, the compositions of the three
phases at equilibrium are fixed at the cor-
ners of the triangle. The relative propor-
tions of the three phases are given by the
lever rule of tie-triangles, which can be de-
rived from mass balance considerations. At
an overall composition qin Fig. 1-26 for
example, the relative proportion of the g
phase is given by projecting a straight line
from the gcorner of the triangle (point c)
through the overall composition qto the
opposite side of the triangle, point p. Then:
(moles of g)/(total moles) =qp/cpif com-
positions are expressed in mole fractions,
or (weight of g)/(total weight) =qp/cpif
compositions are in weight percent.
Isothermal ternary phase diagrams are
generally composed of a number of these
topological units. An example for the Al–
Zn–Mg system at 25 °C is shown in Fig.
1-27 (Köster and Dullenkopf, 1936). The
b, g, d, q, hand zphases are binary inter-
metallic compounds with small (~1 to 6%)
Figure 1-26.A tie-triangle in a ternary isothermal
section illustrating the lever rule and the extension
rule.www.iran-mavad.com
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57. ranges of stoichiometry which can dissolve
a limited amount (~1 to 6%) of the third
component. The tphase is a ternary phase
with a single-phase region existing over a
fairly extensive oval-shaped central com-
position range. Examination of Fig. 1-27
shows that it consists of the topological
units of Fig. 1-26.
An extension rule, a case of Schreine-
makers’ Rule(Schreinemakers, 1915), see
Sec. 1.7.5, for ternary tie-triangles is illus-
trated in Fig. 1-26. At each corner, the
extension of the boundaries of the single-
phase regions, indicated by the broken
lines, must either both project into the tri-
angle as at point a, or must both project
outside the triangle as at point b. Further-
more, the angle between these extensions
must be less than 180°. For a proof, see
Lipson and Wilson (1940) or Pelton
(1995).
Many published phase diagrams violate
this rule. For example, it is violated in Fig.
1-27 at the d-corner of the (e+d+t) tie-tri-
angle.
Another important rule of construction,
whose derivation is evident, is that within
any two-phase region tie-lines must never
cross one another.
1.7.5 Ternary Isopleths
(Constant Composition Sections)
A vertical isopleth, or constant composi-
tion section through the space model of the
Bi–Sn–Cd system, is shown in Fig. 1-28.
The section follows the line ABin Fig.
1-22.
The phase fields on Fig. 1-28 indicate
which phases are present when an alloy
with an overall composition on the line AB
is equilibrated at any temperature. For ex-
ample, consider the cooling, from the liq-
uid state, of an alloy of composition a
which is on the line AB(see Fig. 1-22). At
T≈465 K, precipitation of the solid (Cd)
phase begins at point ain Fig. 1-28. At
T≈435 K (point bin Figs. 1-22 and 1-28)
the solid (Sn) phase begins to appear. Fi-
nally, at the eutectic temperature T
E, the
ternary reaction occurs, leaving solid (Cd)
+ (Bi) + (Sn) at lower temperatures. The
intersection of the isopleth with the univar-
iant lines on Fig. 1-22 occurs at points fand
gwhich are also indicated on Fig. 1-28.
The intersection of this isopleth with the
isothermal section at 423 K is shown in
Fig. 1-25. The points s, t, uand vof Fig.
1-25 are also shown on Fig. 1-28.
It is important to note that on an isopleth
the tie-lines do not, in general, lie in the
plane of the diagram. Therefore, the dia-
gram provides information only on which
phases are present, not on their composi-
tions. The boundary lines on an isopleth do
not in general indicate the phase composi-
tions, only the temperature at which a
phase appears or disappears for a given
overall composition. The lever rule cannot
be applied on an isopleth.
46 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-27.Ternary isothermal section of the Al–
Zn–Mg system at 25°C at P =1 bar (after Köster and
Dullenkopf, 1936). (Reprinted from Pelton, 1983.)www.iran-mavad.com
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58. 1.7 Ternary and Multicomponent Phase Diagrams 47
Certain geometrical rules apply to iso-
pleths. As a phase boundary line is crossed,
one and only one phase either appears or
disappears. This Law of Adjoining Phase
Regions(Palatnik and Landau, 1964) is il-
lustrated by Fig. 1-28. The only apparent
exception occurs for the horizontal invari-
ant line at T
E. However, if we consider this
line to be a degenerate infinitely narrow
four-phase region (L + (Cd) + (Bi) + (Sn)),
then the law is also obeyed here.
Three or four boundary lines meet at in-
tersection points. At an intersection point,
such as point for g, Schreinemakers’ Rule
applies. This is discussed in Sec. 1.9.
Apparent exceptions to these rules (such
as, for example, five boundaries meeting at
an intersection point) can occur if the sec-
tion passes exactly through a node (such
as a ternary eutectic point) of the space
model. However, these apparent exceptions
are really only limiting cases (see Prince,
1963 or 1966).
1.7.5.1 Quasi-Binary Phase Diagrams
Several of the binary phase diagrams
in the preceding sections (Figs. 1-5, 1-10,
1-12, 1-14, 1-18) are actually isopleths
of ternary systems. For example, Fig. 1-12
is an isopleth at constant X
O=n
O/(n
Mg+
n
Ca+n
O) = 0.5 of the Mg–Ca–O system.
However, all tie-lines lie within (or virtu-
ally within) the plane of the diagram be-
cause X
O= 0.5 in every phase. Therefore,
the diagram is called a quasi-binaryphase
diagram.
1.7.6 Multicomponent Phase Diagrams
Only an introduction to multicomponent
phase diagrams will be presented here.
For more detailed treatments see Palatnik
and Landau (1964), Prince (1963), Prince
(1966) and Hillert (1998).
For systems of four or more components,
two-dimensional sections are usually plot-
Figure 1-28.Isopleth (constant composition section) of the Bi–Sn–Cd system at P=1 bar following the line
ABat X
Sn= 0.45 of Fig. 1-22. (Reprinted from Pelton, 1996).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

59. ted with one or more compositional vari-
ables held constant. Hence these sections
are similar to the ternary isopleths dis-
cussed in Sec. 1.7.5. In certain cases, sec-
tions at constant chemical potential of one
or more components (for example, at con-
stant oxygen partial pressure) can be use-
ful. These are discussed in Sec. 1.8.
Two sections of the Fe–Cr–V–C system
(Lee and Lee, 1992) are shown in Figs. 1-
29 and 1-30. The diagram in Fig. 1-29 is a
T-composition section at constant Cr and V
content, while Fig. 1-30 is a section at con-
stant T= 850
o
C and constant C content of
0.3 wt.%. The interpretation and topologi-
cal rules of construction of these sections
are the same as those for ternary isopleths,
as discussed in Sec. 1.7.5. In fact, the same
rules apply to a two-dimensional constant-
composition section for a system of any
number of components. The phase fields
on the diagram indicate the phases present
at equilibrium for an overall composition
lying on the section. Tie-lines do not, in
general, lie in the plane of the diagram, so
the diagram does not provide information
on the compositions or amounts of the
phases present. As a phase boundary is
crossed, one and only one phase appears or
disappears (Law of Adjoining Phase Re-
gions). If temperature is an axis, as in Fig.
1-29, then horizontal invariants like the
line ABin Fig. 1-29 can appear. These can
be considered as degenerate infinitely nar-
row phase fields of (C+ 1) phases, where C
is the number of components (for isobaric
diagrams). For example in Fig. 1-29, on the
line AB, five phases are present. Three or
four phase boundaries meet at intersection
points at which Schreinemakers’ Rule ap-
plies. It is illustrated by the extrapolations
in Fig. 1-29 at points a, band cand in Fig.
1-30 at points b, c, n, iand s(see discus-
sions in Sec. 1.9).
48 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-29.Section of the Fe–Cr–V–C system at 1.5 wt.% Cr and 0.1 wt.% V at P=1 bar (Lee and Lee,
1992).www.iran-mavad.com
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60. 1.7 Ternary and Multicomponent Phase Diagrams 49
1.7.7 Nomenclature for Invariant
Reactions
As discussed in Sec. 1.5.12, in a binary
isobaric temperature–composition phase
diagram there are two possible types of
invariant reactions: “eutectic-type” (bÆa
+g), and “peritectic type” (a+gÆb). In
a ternary system, there are “eutectic-type”
(aÆb+g+d), “peritectic-type” (a+b+
gÆd), and “quasiperitectic-type” (a+b
Æg+d) invariants (Sec. 1.7.3). In a
system of Ccomponents, the number of
types of invariant reaction is equal to C. A
reaction with one reactant, such as
(aÆb+ g+d+e) is clearly a “eutectic-
type” invariant reaction but in general there
is no standard terminology. These reactions
are conveniently described according to the
numbers of reactants and products (in the
direction which occurs upon cooling).
Hence the reaction (a+bÆg+ d+e)is a
2
Æ3 reaction; the reaction (aÆb+g
+d) is a 1
Æ3 reaction; and so on. The ter-
nary peritectic-type 3
Æ1 reaction(a+b
+gÆd) is an invariant reaction in a ter-
nary system, a univariant reaction in a qua-
ternary system, a bivariant reaction in a
quinary system, etc.
1.7.8 Reciprocal Ternary Phase
Diagrams
A reciprocal ternary salt systemis one
consisting of two cations and two anions,
such as the Na
+
, K
+
/F
–
, Cl
–
system of Fig.
1-31. The condition of charge neutrality
(n
Na
++n
K
+=n
F
–+n
Cl
–) removes one degree
of freedom. The system is thus quasiter-
naryand its composition can be repre-
sented by two variables, usually chosen
as the cationic mole fraction X
K=n
K/
(n
Na+n
K) and the anionic mole fraction
X
Cl=n
Cl/(n
F+n
Cl), where n
i= number of
moles of ion i. Note that X
Na=(1–X
K) and
X
F=(1–X
Cl).
Figure 1-30.Section of
the Fe – Cr – V – C system
at 850
o
C and 0.3 wt.% C
at P= 1 bar (Lee and Lee,
1992).www.iran-mavad.com
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61. The assumption has, of course, been
made that the condition (n
Na+n
K=n
F+n
Cl)
holds exactly in every phase. If there is a
deviation from this exact stoichiometry,
then the phase diagram is no longer quasi-
ternary but is an isopleth of the four-com-
ponent Na–K–F–Cl system, and tie-lines
no longer necessarily lie in the plane of the
diagram.
In Fig. 1-31 the cationic and anionic
fractions are plotted as axes of a square.
Compositions corresponding to the four
neutral salts (KF, KCl, NaCl, NaF) are
found at the corners of the square. Edges of
the square correspond to the binary sub-
systems such as NaF–NaCl. A ternary
space model (analogous to Fig. 1-23) can
be constructed with temperature as vertical
axis. The phase diagram of Fig. 1-31 is a
polythermal projection of the liquidus sur-
face upon the composition square.
In this system, three of the binary edges
are simple eutectic systems, while the
NaCl–KCl binary system exhibits a sol-
idus/liquidus minimum. There is a ternary
eutectic at 570 °C in Fig. 1-31(b). The
NaF–KCl diagonal contains a saddle point
at 648 °C in Fig. 1-31(b). This saddle point
is a eutectic of the quasibinary system
NaF–KCl. That is, a binary phase diagram
NaF–KCl could be drawn with one simple
eutectic at 648 °C. However, the NaCl–KF
system, which forms the other diagonal, is
not a quasibinary system. If compositions
lying on this diagonal are cooled at equilib-
rium from the liquid, solid phases whose
compositions do not lie on this diagonal
can precipitate. Hence, a simple binary
phase diagram cannot be drawn for the
NaCl–KF system.
For systems such as Ca
2+
, Na
+
/F
–
, SO
4
2–
in which the ions do not all have the same
charge, composition axes are conveniently
expressed as equivalent ionic fractions
(e.g. Y
Ca=2n
Ca/(2n
Ca+n
Na)), see Sec.
1.9.2.1.
The concept of reciprocal systems can be
generalized beyond simple salt systems
and is closely related to the sublattice
model (Sec. 1.10.1).
50 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-31.Projection of the liquidus surface of
the Na
+
, K
+
/F
–
, Cl
–
reciprocal ternary system.
a) Calculated from optimized binary thermodynamic
parameters.
b) As reported by Polyakov (1940).www.iran-mavad.com
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62. 1.8 Phase Diagrams with Potentials as Axes 51
For further discussion and references,
see Pelton (1988b) and Blander (1964).
1.8 Phase Diagrams
with Potentials as Axes
So far we have considered mainly iso-
baric temperature–composition phase dia-
grams. However there are many other
kinds of phase diagrams of interest in ma-
terials science and technology with pres-
sure, chemical potentials, volume, etc. as
axes. These can be classified into geomet-
rical types according to their rules of con-
struction.
For instance, binary isothermal P–Xdi-
agrams as in Fig. 1-8 are members of the
same type as binary isobaric T–Xdiagrams
because they are both formed from the
same topological units of construction.
Other useful phase diagrams of this same
geometrical type are isothermal chemical
potential–composition diagrams for ter-
nary systems. An example is shown in
the lowest panel of Fig. 1-32 (Pelton and
Thompson, 1975) for the Co–Ni–O
system at T= 1600 K (and at a constant to-
tal hydrostatic pressure of 1 bar). Here the
logarithm of the equilibrium partial pres-
sure of O
2is plotted versus the metal ratio
x=n
Ni/(n
Co+n
Ni), where n
i= number of
moles of i. There are two phases in this
system under these conditions, a solid alloy
solution stable at lower p
O
2
, and a solid so-
lution of CoO and NiO stable at higher p
O
2
.
For instance, point agives p
O
2
for the equi-
librium between pure Co and pure CoO at
1600 K. Between the two single-phase re-
gions is a two-phase (alloy + oxide) region.
At any overall composition on the tie-line
cdbetween points cand d, two phases will
be observed, an alloy of composition dand
an oxide of composition c.The lever rule
applies just as for binary T–Xdiagrams.
The usual isothermal section of the ter-
nary Co–Ni–O system at 1600 K is shown
in the top panel of Fig. 1-32. There are two
single-phase regions with a two-phase re-
gion between them. The single-phase areas
are very narrow because oxygen is only
very slightly soluble in the solid alloy
and CoO and NiO are very stoichiometric
oxides. In the central panel of Fig. 1-32
this same diagram is shown with the com-
position triangle “opened up” by putting
the oxygen corner at infinity. This can be
done if the vertical axis becomes
h=n
O/
(n
Co+n
Ni) with the horizontal axis as
x=n
Ni/(n
Co+n
Ni). These are known as
Jänecke coordinates.It can be seen in Fig.
1-32 that each tie-line, ef,of the isothermal
section corresponds to a tie-line cdof the
Figure 1-32.Corresponding phase diagrams for the
Co – Ni – O system at 1600 K (from Pelton and
Thompson, 1973).www.iran-mavad.com
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63. logp
O
2
–xdiagram. This underscores the
fact that every tie-line of a ternary isother-
mal section corresponds to a constant chem-
ical potential of each of the components.
Another example of a logp
O
2
–xdiagram
is shown for the Fe–Cr–O system at
1573 K in the lower panel of Fig. 1-33
(Pelton and Schmalzried, 1973). The corre-
sponding ternary isothermal section in
Jänecke coordinates is shown in the upper
panel. Each of the invariant three-phase
tie-triangles in the isothermal section
corresponds to an invariant line in the
logp
O
2
–xdiagram. For example, the (spi-
nel + (Fe, Cr) O + alloy) triangle with cor-
ners at points a,band ccorresponds to the
“eutectic-like” or eutecularinvariant with
the same phase compositions a,band cat
logp
O
2
≈–10.7. We can see that within a
three-phase tie-triangle, p
O
2
is constant.
An example of yet another kind of phase
diagram of this same geometrical type is
shown in Fig. 1-34. For the quaternary
Fe–Cr–O
2–SO
2system at T = 1273 K and
at constant p
SO
2
=10
–7
bar, Fig. 1-34 is a
plot of logp
O
2
versus the molar metal ratio
52 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-33.Correspond-
ing phase diagrams for the
Fe–Cr–O system at 1573 K
(Pelton and Schmalzried,
1973). Experimental points
from Katsura and Muan
(1964).www.iran-mavad.com
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64. 1.8 Phase Diagrams with Potentials as Axes 53
x. Since logp
O
2
varies as –1/2 logp
S
2
when
p
SO
2
and Tare constant, Fig. 1-34 is also a
plot of logp
S
2
versus x.
Plotting Tversus
xat constant p
O
2
in the
Fe–Cr–O system, or at constant p
O
2
and
p
SO
2
in the Fe–Cr–SO
2–O
2system, will
also result in phase diagrams of this same
geometrical type. Often for ceramic
systems, we encounter “binary” phase dia-
grams such as that for the “CaO–Fe
2O
3”
system in Fig. 1-35, which has been taken
from Phillips and Muan (1958). How are
we to interpret such a diagram? How, for
instance, do we interpret the composition
Figure 1-34.Calculated phase diagram of log p
O
2
versus molar metal ratio at T= 1273.15 K and p
SO
2
=10
–7
bar
for the Fe–Cr–SO
2–O
2system.
Figure 1-35.Phase diagram for the
“CaO–Fe
2O
3” system in air (p
O
2
=
0.21 bar) from Phillips and Muan
(1958) (Reprinted by permission of the
American Ceramic Society from Levin
et al., 1964).www.iran-mavad.com
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65. axis when applied to the magnetite phase?
In light of the preceding discussion, it can
be seen that such diagrams are really T–
x
plots at constant p
O
2
, where xis the metal
ratio in any phase. The diagram will be dif-
ferent at different oxygen partial pressures.
If p
O
2
is not fixed, the diagram cannot be
interpreted.
It can be seen that the diagrams dis-
cussed above are of the same geometrical
type as binary T–Xdiagrams because they
are all composed of the same geometrical
units of construction as in Fig. 1-16. Their
interpretation is thus immediately clear
to anyone familiar with binary T–Xdia-
grams. Chemical potential–composition
54 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-36.Pressure-temperature phase diagram of
H
2O.
Figure 1-37.Corresponding phase diagrams for the Fe – O system at P
TOTAL= 1 bar (after Muan and Osborn,
1965).www.iran-mavad.com
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66. 1.8 Phase Diagrams with Potentials as Axes 55
diagrams (Figs. 1-32 to 1-34) are useful in
the study of hot corrosion, metallurgical
roasting processes, chemical vapor deposi-
tion, and many aspects of materials pro-
cessing.
Another important geometrical type of
phase diagram is exemplified by P–T
phase diagrams for one-component sys-
tems, as shown for H
2O in Fig. 1-36. In
such diagrams (see also Chapter 10 by
Kunz (2001)) bivariant single-phase re-
gions are indicated by areas, univariant
two-phase regions by lines, and invariant
three-phase regions by triple points.An
important rule of construction is the exten-
sion rule, which is illustrated by the broken
lines in Fig. 1-36. At a triple point, the
extension of any two-phase line must pass
into the single-phase region of the third
phase. Clearly, the predominance diagrams
of Figs. 1-1 to 1-3 are of this same geomet-
rical type.
As yet another example of this geomet-
rical type of diagram, a plot of RTlnp
O
2
versus Tfor the Fe–O system is shown in
Fig. 1-37(b). Again, one-, two- and three-
phase regions are indicated by areas, lines
and triple points respectively. Fig. 1-37(a)
is the binary T–composition phase diagram
for the Fe–O system. The correspondence
between Figs. 1-37(a) and 1-37(b) is evi-
dent. Each two-phase line of Fig. 1-37(b)
“opens up” to a two-phase region of Fig.
1-37(a). Each tie-line of a two-phase re-
gion in Fig. 1-37(a) can thus be seen to
correspond to a constant p
O
2
. Triple points
in Fig. 1-37(b) become horizontal invari-
ant lines in Fig. 1-37(a).
Yet another type of phase diagram is
shown in Fig. 1-38. This is an isothermal
Figure 1-38.Phase diagram of log p
S
2
versus log p
O
2
at 1273 K and constant molar metal ratio n
Cr/(n
Fe+n
Cr) =
0.5 in the Fe – Cr – S
2–O
2system.www.iran-mavad.com
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67. section at constant molar metal ratio
n
Cr/(n
Fe+n
Cr) = 0.5 for the Fe–Cr–S
2–O
2
system. This diagram was calculated ther-
modynamically from model parameters.
The axes are the equilibrium sulfur and
oxygen partial pressures. Three or four
boundary lines can meet at an intersection
point. Some of the boundary lines on Fig.
1-38 separate a two-phase region (a+b)
from another two-phase region (a+g).
These lines thus represent the conditions
for three-phase (a+b+g) equilibrium.
1.9 General Phase Diagram
Geometry
Although the various phase diagrams
shown in the preceding sections may ap-
pear to have quite different geometries, it
can be shown that, in fact, all true phase di-
agram sections obey the same set of geo-
metrical rules. Although these rules do not
apply directly to phase diagram projections
such as Figs. 1-22, 1-24 and 1-31, such di-
agrams can be considered to consist of por-
tions of several phase diagram sections
projected onto a common plane.
By “true” phase diagram we mean one in
which each point of the diagram represents
one unique equilibrium state. In the present
section we give the general geometrical
rules that apply to all true phase diagram
sections, and we discuss the choices of
axes and constants that ensure that the dia-
gram is a true diagram.
We must first make some definitions. In
a system of Ccomponents we can define
(C+ 2) thermodynamic potentials
f
i. These
are T,P,
m
1, m
2,…,m
C, where m
iis the
chemical potential defined in Eq. (1-23).
For each potential there is a corresponding
extensive variable q
irelated by:
f
i= (∂U/∂q
i)
q
j
(j≠i) (1-114)
where Uis the internal energy of the
system. The corresponding potentials and
extensive variables are listed in Table 1-1.
It may also be noted that the corresponding
pairs are found together in the terms of the
general Gibbs–Duhem equation:
SdT–VdP+
Sn
idm
i= 0 (1-115)
1.9.1 General Geometrical Rules
for All True Phase Diagram Sections
The Law of Adjoining Phase Regionsap-
plies to all true sections. As a phase boun-
dary line is crossed, one and only one
phase either appears or disappears.
If the vertical axis is a potential (T,P,
m
i), then horizontal invariant lines like the
eutectic line in Fig. 1-12 or the line ABin
Fig. 1-29 will be seen when the maximum
number of phases permitted by the phase
rule are at equilibrium. However, if these
are considered to be degenerate infinitely
narrow phase fields, then the Law of Ad-
joining Phase Regions still applies. This is
illustrated schematically in Fig. 1-39 where
the three-phase eutectic line has been
“opened up”. Similarly, if both axes are po-
tentials, then many phase boundaries may
be degenerate infinitely narrow regions.
For example, all phase boundaries on Figs.
1-1 to 1-3, 1-36 and 1-37(b) are degenerate
two-phase regions which are schematically
shown “opened up” on Fig. 1-40.
All phase boundary lines in a true phase
diagram meet at nodes where exactly four
lines converge, as in Fig. 1-41. Nphases
(a
1, a
2, …, a
N) where N≥1 are common
56 1 Thermodynamics and Phase Diagrams of Materials
Table 1-1.Corresponding pairs of potentials f
iand
extensive variables q
i.
f
i:TP m
1 m
2…m
C
q
i: S –Vn
1 n
2…n
Cwww.iran-mavad.com
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68. 1.9 General Phase Diagram Geometry 57
to all four regions. Schreinemakers’ Rule
states that the extensions of the boundaries
of the N -phase region must either both lie
within the (N +1)-phase regions as in Fig.
1-41 or they must both lie within the
(N+ 2)-phase region. This rule is illustrated
by the extrapolations in Fig. 1-29 at points
a, band cand in Fig. 1-30 at points b, c, n,
iand s. The applicability of Schreine-
makers’ Rule to systems of any number of
components was noted by Hillert (1985)
and proved by Pelton (1995). In the case of
degenerate phase regions, all nodes can
still be considered to involve exactly four
boundary lines if the degenerate boundar-
ies are “opened up” as in Figs. 1-39 and
1-40.
An objection might be raised that a
minimum or a maximum in a two-phase re-
gion in a binary temperature–composition
phase diagram, as in Fig. 1-10 or in the
lower panel of Fig. 1-16, represents an ex-
Figure 1-39.An isobaric binary T – X
phase diagram (like Fig. 1-12) with the
eutectic line “opened up” to illustrate that
this is a degenerate 3-phase region.
Figure 1-40.A potential–potential phase diagram
(like Fig. 1-1 or Fig. 1-36) with the phase boundaries “opened up” to illustrate that they are degenerate 2- phase regions.
Figure 1-41.A node in a true phase diagram sec-
tion.www.iran-mavad.com
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69. ception to Schreinemakers’ Rule. How-
ever, the extremum in such a case is not ac-
tually a node where four phase boundaries
converge, but rather a point where two
boundaries touch. Such extrema in which
two phase boundaries touch with zero
slope may occur for a C-phase region in a
phase diagram of a C-component system
when one axis is a potential. For example,
in an isobaric temperature–composition
phase diagram of a four-component sys-
tem, we may observe a maximum or a min-
imum in a four-phase region separating two
three-phase regions. A similar maximum
or minimum in a (C –n)-phase region,
where n> 0, may also occur, but only for
a degenerate or special composition path.
For further discussion, see Hillert (1998).
1.9.1.1 Zero Phase Fraction Lines
All phase boundaries on true phase dia-
gram sections are zero phase fraction
(ZPF) lines, a very useful concept intro-
duced by Gupta et al. (1986). There are
ZPF lines associated with each phase. On
one side of its ZPF line the phase occurs,
while on the other side it does not. For ex-
ample, in Fig. 1-30 the ZPF line for the a
phase is the line abcdef. The ZPF line for
the gphase is g hijkl. For the MC phase the
ZPF line is mnciopq . The ZPF line for
M
7C
3is rnbhspket, and for M
23C
6it is
udjosv. These five ZPF lines yield the en-
tire two-dimensional phase diagram. Phase
diagram sections plotted on triangular co-
ordinates as in Figs. 1-25 and 1-27 also
consist of ZPF lines.
In the case of phase diagrams with de-
generate regions, ZPF lines for two differ-
ent phases may be coincident over part
of their lengths. For example, in Fig. 1-12,
line CABDis the ZPF line of the liquid,
while CEBFand DEAGare the ZPF lines
for the aand bphases respectively. In
Fig. 1-1, all lines are actually two coinci-
dent ZPF lines.
The ZPF line concept is very useful in
the development of general algorithms for
the thermodynamic calculation of phase di-
agrams as discussed in Sec. 1-12.
1.9.2 Choice of Axes and Constants
of True Phase Diagrams
In a system of C components, a two-
dimensional diagram is obtained by choos-
ing two axis variables and holding (C–1)
other variables constant. However, not all
choices of variables will result in a true
phase diagram. For example on the P–V
diagram for H
2O shown schematically in
Fig. 1-42, at any point in the area where the
(S + L) and (L + G) regions overlap there
are two possible equilibrium states of the
system. Similarly, the diagram of carbon
activity versus X
Crat constant Tand Pin
the Fe–Cr–C system in Fig. 1-43 (Hillert,
1997) exhibits a region in which there is no
unique equilibrium state.
In order to be sure that a diagram is a
true phase diagram, we must choose one
and only onevariable (either
f
ior q
i) from
58 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-42.Schematic P–Vdiagram for H
2O. This
is not a true phase diagram.www.iran-mavad.com
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70. 1.9 General Phase Diagram Geometry 59
each of the (C+ 2) conjugate pairs in Table
1-1. (Also, at least one of these must be an
extensive variable q
i.) From among the
n(1≤n≤C+ 2) selected extensive vari-
ables, (n –1) independent ratios are then
formed. These (n –1) ratios along with the
(C+2–n) selected potentials are the (C+1)
required variables. Two are chosen as axis
variables and the remainder are held con-
stant.
As a first example, consider a binary
system with components A–B. The conju-
gate pairs are (T,S), (P,–V), (
m
A, n
A) and
(
m
B,n
B). Let us choose one variable from
each pair as follows: T,P, n
A, n
B. From the
selected extensive variables, n
Aand n
B, we
form a ratio such as n
B/(n
A+n
B)=X
B. The
resultant phase diagram variables are T,P,
X
B. Choosing any two as axes and holding
the third constant will give a true phase di-
agram as in Fig. 1-6 or Fig. 1-8.
As a second example, consider Fig. 1-38
for the Fe–Cr–S
2–O
2system. We choose
one variable from each conjugate pair as
follows: T, P,
m
S
2
, m
O
2
, n
Fe, n
Cr. From the
selected extensive variables we form the
ratio n
Fe/(n
Fe+n
Cr). Fig. 1-38 is a plot
of
m
S
2
versus m
O
2
at constant T , Pand
n
Fe/(n
Fe+n
Cr).
In Fig. 1-28 the selected variables are T,
P,n
Bi, n
Snand n
Cd, and ratios are formed
from the selected extensive variables as
n
Cd/(n
Cd+n
Bi) and n
Sn/(n
Cd+n
Bi+n
Sn)
=X
Sn. Fig. 1-28 is a plot of Tversus
n
Cd/(n
Cd+n
Bi) at constant Pand X
Sn.
Fig. 1-42, the P–Vdiagram for H
2O, is
not a true phase diagram because Pand V
are members of the same conjugate pair.
For the diagram shown in Fig. 1-43, we can
choose one variable from each pair as fol-
lows: T,P,
m
C,n
Fe, n
Cr.However the verti-
cal axis is X
Cr=n
Cr/(n
Fe+n
Cr+n
C). This ra-
tio is not allowed because it contains n
C
which is not on the list of chosen variables.
That is, since we have chosen
m
Cto be an
axis variable, we cannot also choose n
C.
Hence, Fig. 1-43 is not a true phase dia-
gram. A permissible choice for the vertical
axis would be n
Cr/(n
Fe+n
Cr) (see Fig.
1-33). Note that many regions of Figs. 1-42
and 1-43 do represent unique equilibrium
states. That is, the procedure given here is a
sufficient, but not necessary, condition for
constructing true phase diagrams.
To apply this procedure simply, the com-
ponents of the system should be formally
defined to correspond to the desired axis
variables or constants. For example, in Fig.
1-1 we wish to plot p
SO
2
and logp
O
2
as
axes. Hence we define the components as
Cu–SO
2–O
2rather than Cu–S–O.
In several of the phase diagrams in this
chapter, logp
ior RTln p
ihas been substi-
tuted for
m
ias axis variable or constant.
From Eq. (1-32), this substitution can
clearly be made if Tis constant. However,
even when Tis an axis of the phase dia-
gram as in Fig. 1-37(b), this substitution is
still permissible since
m
i
0is a monotonic
function of T. The substitution of lna
ifor
Figure 1-43.Carbon activity versus mole fraction
of Cr at constant Tand Pin the Fe–Cr–C system.
This is not a true phase diagram (from Hillert, 1997).www.iran-mavad.com
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71. m
iresults in a progressive expansion and
displacement of the axis with increasing T
that preserves the overall geometry of the
diagram.
1.9.2.1 Tie-lines
If only potentials (T,P,
m
i) are held con-
stant, then all tie-lines lie in the plane of
the phase diagram section. In this case, the
compositions of the individual phases at
equilibrium can be read from the phase
diagram, and the lever rule applies as, for
example, in Figs. 1-6, 1-25, 1-33 or 1-34.
However, if a ratio of extensive variables,
such as a composition, is held constant as
in the isopleths of Figs. 1-28 to 1-30, then
in general, tie-lines do not lie in the plane.
If both axes are composition variables
(ratios of n
i), and if only potentials are held
constant, then it is desirable that the tie-
lines (which lie in the plane) be straight
lines. It can be shown (Pelton and Thomp-
son, 1975) that this will only be the case if
the denominators of the two composition
variable ratios are the same. For example,
in the central panel of Fig. 1-32, which is in
Jänecke coordinates, the composition vari-
ables, n
Co/(n
Co+n
Ni) and n
O/(n
Co+n
Ni),
have the same denominator. This same dia-
gram can be plotted on triangular coordi-
nates as in the upper panel of Fig. 1-32 and
such a diagram can also be shown (Pelton
and Thompson, 1975) to give straight tie-
lines.
Similarly, in the quasiternary reciprocal
phase diagram of Fig. 1-31 the vertical and
horizontal axes are n
Na/(n
Na+n
K) and
n
Cl/(n
Cl+n
F). To preserve charge neutral-
ity, (n
Na+n
K)=(n
Cl+n
F), and so the tie-
lines are straight. Generally, in quasiter-
nary reciprocal salt phase diagrams,
straight tie-lines are obtained by basing the
composition on one equivalent of charge.
For example, in the CaCl
2–NaCl–CaO–
Na
2O system we would choose as axes
the equivalent cationic and anionic frac-
tions, n
Na/(n
Na+2n
Ca) and n
Cl/(n
Cl+2n
O),
whose denominators are equal because of
charge neutrality.
1.9.2.2 Corresponding Phase Diagrams
When only potentials are held constant
and when both axes are also potentials,
then the geometry exemplified by Figs. 1-1
to 1-3, 1-26 and 1-37(b) results. Such dia-
grams were called “type-1 phase diagrams”
by Pelton and Schmalzried (1973). If only
potentials are held constant and one axis is
a potential while the other is a composition
variable, then the geometry exemplified
by Figs. 1-8, 1-12, 1-34, 1-37(a), and the
lower panel of Fig. 1-33 results. These
were termed “type-2” diagrams. Finally, if
only potentials are held constant and both
axes are compositions, then a “type-3” dia-
gram as in the upper panels of Figs. 1-32
and 1-33 results.
If the
f
iaxis of a phase diagram is re-
placed by a composition variable that var-
ies as its conjugate variable q
i(ex: q
i/q
j,
q
i/(q
i+q
j)), then the new diagram and the
original diagram are said to form a pair of
corresponding phase diagrams.For in-
stance, Figs. 1-37(a) and 1-37(b) are cor-
responding type-1 and type-2 phase dia-
grams, while Fig. 1-33 shows a corre-
sponding pair of type-2 and type-3 dia-
grams. It is useful to draw corresponding
diagrams beside each other as in Figs. 1-37
or 1-33 because the information contained
in the two diagrams is complementary.
1.9.2.3 Theoretical Considerations
A complete rigorous proof that the pro-
cedure described in this section will always
generate a true phase diagram is beyond
the scope of this chapter. As an outline of
the proof, we start with the generalized
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72. 1.9 General Phase Diagram Geometry 61
stability criterion:
(1-116)
(∂
f
1/∂q
1)
f
2, f
3,…,q
N,q
N+1,q
N+2,…,q
C+2
≥0
This equation states that a potential
f
ial-
ways increases as its conjugate variable q
i
increases when either f
jor q
jfrom every
other conjugate pair is held constant. For
instance,
m
iof a component always in-
creases as that component is added to a
system (that is, as n
iis increased) at con-
stant Tand P, when either the number of
moles or the chemical potential of every
other component is held constant. In a bi-
nary system, for example, this means that
the equilibrium Gibbs energy envelope is
always convex, as shown in Fig. 1-6. If the
envelope were concave, then the system
would be unstable and would separate into
two phases, as shown in Fig. 1-11.
Consider first a phase diagram with axes
f
1and f
2with f
3, f
4,…,f
C+1and q
C+2
constant. Such a diagram is always a true
phase diagram. If the potential
f
1is now re-
placed by q
1, the diagram still remains a
true phase diagram because of Eq. (1-116).
The sequence of equilibrium states that oc-
curs as q
1is increased will be the same as
that which occurs as
f
1is increased when
all the other variables (
f
ior q
i) are held
constant.
A true phase diagram is therefore ob-
tained if the axis variables and constants
are chosen from the variables
f
1, f
2,…,
f
N, q
N+1, q
N+2,…,q
C+1with q
C+2held
constant. The extensive variables can be
normalized as (q
i/q
C+2) or by any other in-
dependent and unique set of ratios.
It should be noted that at least one exten-
sive variable, q
C+2, is considered to be con-
stant across the entire diagram. In practice,
this means that one of the extensive vari-
ables must be either positive or negative
everywhere on the diagram. For certain
formal choices of components, extensive
composition variables can have negative
values. For example, in the predominance
diagram of Fig. 1-1, if the components are
chosen as Cu–SO
2–O
2, then the com-
pound Cu
2S is written as Cu
2(SO
2)O
–2;
that is n
O
2
= –1. This is no problem in Fig.
1-1, since
m
O
2
rather than n
O
2
was chosen
from the conjugate pair and is plotted as an
axis variable. However, suppose we wish
to plot a diagram of
m
Cuversus m
SO
2
at con-
stant Tand Pin this system. In this case,
the chosen variables would be T,P,
m
SO
2
,
n
O
2
. Since one of the selected extensive
variables must always be positive, and
since n
O
2
is the only selected extensive var-
iable, it is necessary that n
O
2
be positive
everywhere. For instance, a phase field for
Cu
2S is not permitted. In other words, only
compositions in the Cu–SO
2–O
2subsys-
tem are permitted. A different phase dia-
gram would result if we plotted
m
Cuversus
m
SO
2
in the Cu–SO
2–S
2subsystem with n
S
2
always positive. Cu
2O would then not ap-
pear, for example. That is, at a given
m
Cu
and p
SO
2
we could have a low p
O
2
and a
high p
S
2
in equilibrium with, for example,
Cu
2S, or we could have a high p
O
2
and a
low p
S
2
in equilibrium with, for example,
Cu
2O. Hence the diagram will not be a true
diagram unless compositions are limited to
the Cu–SO
2–O
2or Cu–SO
2–S
2triangles.
As a second example, if
m
SiOand m
COare
chosen as variables in the SiO–CO–O
system, then the diagram must be limited
to n
O> 0 (SiO–CO–O subsystem) or to
n
O< 0 (SiO–CO–Si–C subsystem).
1.9.2.4 Other Sets of Conjugate Pairs
The set of conjugate pairs in Table 1-1 is
only one of many such sets. For example, if
we make the substitution (H=TS+
Sn
im
i)
in Eq. (1-115), then we obtain another form
of the general Gibbs–Duhem equation:
–Hd(1/T) – (V/T)dP+
Sn
id(m
i/T) = 0
(1-117)www.iran-mavad.com
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73. This defines another set of pairs of conju-
gate potentials and extensive variables:
(1/T,–H), (P,–V/T), (
m
i/T,n
i). Choosing
one and only one variable from each pair,
we can construct a true phase diagram by
the procedure described above. However,
these diagrams may be of limited practical
utility. This is discussed by Hillert (1997).
1.10 Solution Models
In Sec. 1.4.7, the thermodynamic expres-
sions for simple ideal substitutional solu-
tions were derived and in Secs. 1.5.7 and
1.6.2, the regular solution model and poly-
nomial extensions thereof were discussed.
For other types of solutions such as ionic
mixtures, interstitial solutions, polymeric
solutions, etc., the most convenient defini-
tion of ideality may be different. In the
present section we examine some of these
solutions. We also discuss structural order-
ing and its effect on the phase diagram. For
further discussion, see Pelton (1997).
1.10.1 Sublattice Models
The sublattice concept has proved to be
very useful in thermodynamic modeling.
Sublattice models, which were first devel-
oped extensively for molten salt solutions,
find application in ceramic, interstitial so-
lutions, intermetallic solutions, etc.
1.10.1.1 All Sublattices Except One
Occupied by Only One Species
In the simplest limiting case, only one
sublattice is occupied by more than one
species. For example, liquid and solid
MgO–CaO solutions can be modeled by
assuming an anionic sublattice occupied
only by O
2–
ions, while Mg
2+
and Ca
2+
ions mix on a cationic sublattice. In this
case the model is formally the same as that
of a simple substitutional solution, because
the site fractions X
Mgand X
Caof Mg
2+
and
Ca
2+
cations on the cationic sublattice are
numerically equal to the overall component
mole fractions X
MgOand X
CaO. Solid and liq-
uid MgO–CaO solutions have been shown
(Wu et al., 1993) to be well represented by
simple polynomial equations for g
E
.
As a second example, the intermetallic e-
FeSb phase exhibits non-stoichiometry to-
ward excess Fe. This phase was modeled
(Pei et al., 1995) as a solution of Fe and
stoichiometric FeSb by assuming two sub-
lattices: an “Fe sublattice” occupied only
by Fe atoms and an “Sb sublattice” occu-
pied by both Fe and Sb atoms such that, per
gram atom,
Dg
m= 0.5RT(y
Felny
Fe+ y
Sblny
Sb)
+ay
Fey
Sb (1-118)
where y
Sb=(1–y
Fe)=2X
Sbis the site frac-
tion of Sb atoms on the “Sb sites” and ais
an empirical polynomial in y
Sb.
1.10.1.2 Ionic Solutions
Let us take as an example a solution,
solid or liquid, of NaF, KF, NaCl and KCl
as introduced in Sec. 1.7.8. If the cations
are assumed to mix randomly on a cationic
sublattice while the anions mix randomly
on an anionic sublattice, then the molar
Gibbs energy of the solution can be mod-
eled by the following equation which con-
tains an ideal mixing term for each sublat-
tice:
g= (X
NaX
Clg
0
NaCl
+ X
KX
Fg
0
KF
+ X
NaX
Fg
0
NaF
+ X
KX
Clg
0
KCl
) (1-119)
+ RT(X
NalnX
Na+ X
KlnX
K)
+ RT(X
FlnX
F+ X
CllnX
Cl) + g
E
where the factor (X
NaX
Cl), for example, is
the probability, in a random mixture, of
62 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
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74. 1.10 Solution Models 63
finding a Na ion and a Cl ion as nearest
neighbors. Differentiation of Eq. (1-119)
gives the following expression for the ac-
tivity of NaF:
RTlna
NaF= –X
KX
ClDG
exchange
(1-120)
+ RTln (X
NaX
F) + g
E
NaF
where DG
exchange
is the Gibbs energy
change for the following exchange reaction
among the pure salts:
NaCl + KF = NaF + KCl; (1-121)
DG
exchange
= g
0
NaF
+ g
0
KCl
– g
0
NaCl
– g
0
KF
In this example, DG
exchange
< 0. The salts
NaF and KCl are thus said to form the
stable pair.The first term on the right of
Eq. (1-120) is positive. The members of the
stable pair thus exhibit positive deviations,
and in Fig. 1-31 this is reflected by the flat
liquidus surfaces with widely spaced iso-
therms for NaF and KCl. That is, the mix-
ing of pure NaF and KCl is unfavorable be-
cause it involves the formation of K
+
–F
–
and Na
+
–Cl
–
nearest-neighbor pairs at the
expense of the energetically preferable
Na
+
–F
–
and K
+
–Cl
–
pairs. If DG
exchange
is
sufficiently large, a miscibility gap will be
formed, centered close to the stable diago-
naljoining the stable pair.
Blander (1964) proposed the following
expression for g
E
in Eq. (1-119):
g
E
= X
Nag
E
NaCl–NaF
+ X
Kg
E
KCl–KF
(1-122)
+ X
Fg
E
NaF–KF
+ X
Clg
E
NaCl–KCl
– X
NaX
KX
FX
Cl(DG
exchange
)
2
/ZRT
where, for example, g
E
NaCl–NaF
is the excess
Gibbs energy in the NaCl–NaF binary
system at the same cationic fraction X
Naas
in the ternary, and where Zis the first coor-
dination number. That is, g
E
contains a
contribution from each binary system. The
final term in Eq. (1-122) is a first-order
correction for non-random mixing which
accounts for the fact that the number
of Na
+
–F
–
and K
+
–Cl
–
nearest-neighbor
pairs will be higher than the number of
such pairs in a random mixture. This term
is usually not negligible.
The phase diagram in Fig. 1-31(a) was
calculated by means of Eqs. (1-119) and
(1-122) solely from optimized excess
Gibbs energies of the binary systems and
from compiled data for the pure salts.
Agreement with the measured diagram is
very good.
Eqs. (1-119) and (1-122) can easily be
modified for solutions in which the num-
bers of sites on the two sublattices are not
equal, as in MgCl
2–MgF
2–CaCl
2–CaF
2
solutions. Also, in liquid salt solutions the
ratio of the number of lattice sites on one
sublattice to that on the other sublattice
can vary with concentration, as in molten
NaCl–MgCl
2–NaF–MgF
2solutions. In
this case, it has been proposed (Saboungi
and Blander, 1975) that the molar ionic
fractions in all but the random mixing
terms of these equations should be replaced
by equivalent ionic fractions. Finally, the
equations can be extended to multicompo-
nent solutions. These extensions are all dis-
cussed by Pelton (1988b).
For solutions such as liquid NaF–KF–
NaCl–KCl for which DG
exchange
is not too
large, these equations are often sufficient.
For solutions with larger exchange Gibbs
energies, however, in which liquid immis-
cibility appears, these equations are gener-
ally unsatisfactory. It was recognized by
Saboungi and Blander (1974) that this is
due to the effect of non-randomness upon
the second nearest-neighbor cation–cation
and anion–anion interactions. To take
account of this, Blander proposed addi-
tional terms in Eq.(1-122). Dessureault and
Pelton (1991) modified Eqs. (1-119) and
(1-122) to account more rigorously for
non-random mixing effects, with good re-www.iran-mavad.com
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75. sults for several molten salt systems with
miscibility gaps. See also section 1.10.4.
1.10.1.3 Interstitial Solutions
As an example of the application of the
sublattice model to interstitial solutions we
will take the f.c.c. phase of the Fe–V–C
system. Lee and Lee (1991) have modeled
this solution using two sublattices: a metal-
lic sublattice containing Fe and V atoms,
and an interstitial lattice containing C
atoms and vacancies, va. The numbers of
sites on each sublattice are equal. An equa-
tion identical to Eq. (1-119) can be written
for the molar Gibbs energy:
g= (X
FeX
vag
0
Feva
+ X
FeX
Cg
0
FeC
+ X
VX
vag
0
Vva
+ X
VX
Cg
0
VC
) (1-123)
+ RT(X
FelnX
Fe+ X
VlnX
V)
+ RT(X
ClnX
C+ X
valnX
va) + g
E
where X
Fe=(1–X
V) and X
C=(1–X
va) are
the site fractions on the two sublattices and
“Feva” and “Vva” are simply pure Fe and
V, i.e., g
0
Feva
=g
0
Fe
. An expression for g
E
as
in Eq. (1-122), although without the final
non-random mixing term, was used by Lee
and Lee with optimized binary g
E
parame-
ters. Their calculated Fe–V–C phase dia-
gram is in good agreement with experimen-
tal data. The sublattice model has been
similarly applied to many interstitial solu-
tions by several authors.
1.10.1.4 Ceramic Solutions
Many ceramic solutions contain two or
more cationic sublattices. As an example,
consider a solution of Ti
2O
3in FeTiO
3(il-
menite) under reducing conditions. There
are two cationic sublattices, the A and B
sublattices. In FeTiO
3, Fe
2+
ions and Ti
4+
ions occupy the A and B sublattices, re-
spectively. With additions of Ti
2O
3, Ti
3+
ions occupy both sublattices. The solu-
tion can be represented as (Fe
2+
1–x
Ti
x
3+)
A
(Ti
4+
1–x
Ti
x
3+)
Bwhere xis the overall mole
fraction of Ti
2O
3. The ions are assumed to
mix randomly on each sublattice so that:
Ds
ideal
= – 2R[(1 –x)ln(1–x) + xlnx]
(1-124)
Deviations from ideal mixing are as-
sumed to occur due to interlattice cation–
cation interactions according to
(Fe
A
2+– Ti
B
4+) + (Ti
A
3+– Ti
B
3+)
= (Fe
A
2+– Ti
B
3+) + (Ti
A
3+– Ti
B
4+)
DG= a+ bT (1-125)
where aand bare the adjustable parameters
of the model. The probability that an A–B
pair is an (Fe
A
2+–Ti
B
3+) or a (Ti
A
3+–Ti
B
4+)
pair is equal to x(1 –x). Hence, g
E
=
x(1–x)(a+bT).
Similar models can be proposed for
other ceramic solutions such as spinels,
pseudobrookites, etc. These models can
rapidly become very complex mathemati-
cally as the number of possible species on
the lattices increases. For instance, in
Fe
3O
4–Co
3O
4spinel solutions, Fe
2+
, Fe
3+
,
Co
2+
and Co
3+
ions are all distributed over
both the tetrahedral and octahedral sublat-
tices. Four independent equilibrium con-
stants are required (Pelton et al., 1979) to
describe the cation distribution even for the
ideal mixing approximation. This com-
plexity has been rendered much more tract-
able by the “compound energy model”
(Sundman and Ågren, 1981; Hillert et al.,
1988). This is not actually a model, but is
rather a mathematical formalism permit-
ting the formulation of various models in
terms of the Gibbs energies, g
0
, of “pseu-
docomponents” so that equations similar to
Eq. (1-119) can be used directly.
64 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
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76. 1.10 Solution Models 65
1.10.1.5 The Compound Energy
Formalism
As an example, the model for the
FeTiO
3–Ti
2O
3solution in Sec. 1.10.1.4
will be reformulated. By taking all com-
binations of an A-sublattice species and
a B-sublattice species, we define two
real components, (Fe
2+
)
A(Ti
4+
)
BO
3and
(Ti
3+
)
A(Ti
3+
)
BO
3, as well as two “pseudo-
components”, (Fe
2+
)
A(Ti
3+
)
BO
3
–and
(Ti
3+
)
A(Ti
4+
)
BO
3
+.
Pseudocomponents, as in the present ex-
ample, may be charged. Similarly to Eq.
(1-119) the molar Gibbs energy can be
written
g= (1–x)
2
g
0
FeTiO
3
+ x
2
g
0
Ti
2O
3
(1-126)
+ x(1–x)g
0
FeTiO
3
–
+ x(1–x)g
0
Ti
2O
3
+
– TDs
ideal
Note that charge neutrality is maintained in
Eq. (1-126). The Gibbs energies of the two
pseudocomponents are calculated from the
equation
DG= a+ bT
= g
0
FeTiO
3
–
+ g
0
Ti
2O
3
+
– g
0
FeTiO
3
– g
0
Ti
2O
3
(1-127)
where DGis the Gibbs energy change of
Eq. (1-125) and is a parameter of the
model. One of g
0
FeTiO
3
–
or g
0
Ti
2O
3
+
may be as-
signed an arbitrary value. The other is then
given by Eq. (1-127). By substitution of
Eq. (1-127) into Eq. (1-126) it may be
shown that this formulation is identical to
the regular solution formulation given in
Sec. 1.10.1.4. Note that excess terms, g
E
,
could be added to Eq. (1-126), thereby giv-
ing more flexibility to the model. In the
present example, however, this was not re-
quired.
The compound energy formalism is de-
scribed and developed by Barry et al.
(1992), who give many more examples. An
advantage of formulating the sublattice
model in terms of the compound energy
formalism is that it is easily extended to
multicomponent solutions. It also provides
a conceptual framework for treating many
different phases with different structures.
This facilitates the development of com-
puter software and databases because many
different types of solutions can be treated
as cases of one general formalism.
1.10.1.6 Non-Stoichiometric Compounds
Non-stoichiometric compounds are gen-
erally treated by a sublattice model. Con-
sider such a compound A
1–dB
1+d. The sub-
lattices normally occupied by A and B at-
oms will be called, respectively, the A-sub-
lattice and the B-sublattice. Deviations
from stoichiometry (where
d= 0) can occur
by the formation of defects such as B atoms
on A sites, vacant sites, atoms occupying
interstitial sites, etc. Generally, one type of
defect will predominate for solutions with
excess A and another type will predomi-
nate for solutions with excess B. These are
called the majority defects.
Consider first a solution in which the
majority defects are A atoms on B sites and
B atoms on A sites: (A
1–xB
x)
A(A
yB
1–y)
B. It
follows that
d=(x–y). In the compound
energy formalism we can write, for the mo-
lar Gibbs energy,
g= (1–x)(1–y )g
0
AB
+ (1–x)yg
0
AA
+ x(1–y)g
0
BB
+ xyg
0
BA
(1-128)
+ RT[xlnx+ (1–x) ln (1–x)
+ ylny+ (1–y) ln (1–y)]
where g
0
AB
is the molar Gibbs energy of
(hypothetical) defect-free stoichiometric
AB. Now the defect concentrations at equi-
librium are those that minimize g. There-
fore, setting (∂g/∂x)=(∂g/∂y) = 0 at con-
stant
d, we obtain
(1-129)
xy
xy
xy
RT()()
exp ( )
11
1
12
−−
−−−
+⎛
⎝
⎜
⎞
⎠
⎟=
DDggwww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

77. where Dg
1=(g
0
AA
–g
0
AB
) and D g
2=
(g
0
BB
–g
0
AB
) are the Gibbs energies of for-
mation of the majority defects and where
g
0
BA
has been set equal to g
0
AB
. At a given
composition
d=(x–y), and for given val-
ues of the parameters Dg
1and Dg
2, Eq.
(1-129) can be solved to give xand y,
which can then be substituted into Eq.
(1-128) to give g. The more positive are
Dg
1and Dg
2, the more steeply grises on ei-
ther side of its minimum, and the narrower
is the range of stoichiometry of the com-
pound.
Consider another model in which the
majority defects are vacancies on the B-
sublattice and B atoms on interstitial sites.
We now have three sublattices with occu-
pancies (A)
A(B
1–yva
y)
B(B
xva
1–x)
Iwhere
“I” indicates the interstitial sublattice. The
A-sublattice is occupied exclusively by A
atoms. A vacancy is indicated by va. Stoi-
chiometric defect-free AB is represented
by (A) (B) (va) and (x–y)=2
d/(1–d).
Per mole of A
1–dB
1+d, the Gibbs energy is:
g= (1–
d) {[(1–x)(1–y)g
0
ABva
+ (1–x)yg
0
Av a v a
+ x(1–y)g
0
ABB
+ xyg
0
AvaB
+ RT[xlnx+ (1–x) ln (1–x)
+ ylny+ (1–y) ln (1–y)]} (1-130)
Eq. (1-130) is identical to Eq. (1-128) apart
from the factor (1–
d), and gives rise to an
equilibrium constant as in Eq. (1-129).
Other choices of majority defects result in
very similar expressions. The model can
easily be modified to account for other
stoichiometries A
mB
n, for different num-
bers of available interstitial sites, etc., and
its extension to multicomponent solutions
is straightforward.
1.10.2 Polymer Solutions
For solutions of polymers in monomeric
solvents, very large deviations from simple
Raoultian ideal behaviour (i.e. from Eq. (1-
40)) are observed. This large discrepancy
can be attributed to the fact that the indi-
vidual segments of the polymer molecule
have considerable freedom of movement.
Flory (1941, 1942) and Huggins (1942)
proposed a model in which the polymer
segments are distributed on the solvent
sites. A large polymer molecule can thus be
oriented (i.e. bent) in many ways, thereby
greatly increasing the entropy. To a first ap-
proximation the model gives an ideal mix-
ing term with mole fractions replaced by
volume fractions in Eq. (1-45):
Lewis and Randall (1961) have com-
pared the Flory–Huggins equation with
experimental data in several solutions.
In general, the measured activities lie be-
tween those predicted by Eq. (1-131) and
by the Raoultian ideal equation, Eq. (1-45).
A recent review of the thermodynamics
of polymer solutions is given by Trusler
(1999).
1.10.3 Calculation of Limiting Slopes
of Phase Boundaries
From the measured limiting slope
(dT/dX)
X
A=1of the liquidus at the melting
point of a pure component A, much infor-
mation about the extent of solid solubility,
as well as the structure of the liquid, can be
inferred. Similar information can be ob-
tained from the limiting slopes of phase
boundaries at solid-state transformation
points of pure components.
Eq. (1-65) relates the activities along the
liquidus and the solidus to the Gibbs en-
ergy of fusion:
RTlna
l
A
– RTlna
s
A
= – Dg
0
f(A)
(1-132)
Dg
v
vv
v
vv
m
ideal
A
AA
0
AA
0
BB
0
B
BB
0
AA
0
BA
0=
(1-131)
RT X
X
XX
X
X
XX
ln
ln
+
⎛
⎝
⎜
+
+
⎞
⎠
⎟
66 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

78. 1.10 Solution Models 67
In the limit X
AÆ1, the liquidus and solidus
converge at the melting point T
f(A). Let us
assume that, in the limit, Raoult’s Law, Eq.
(1-40), holds for both phases. That is,
a
l
A
=X
l
A
and a
s
A
=X
s
A
. Furthermore, from
Eq. (1-60),
Dg
0
f (A)
ÆDh
f(A)(1 –T/T
f(A))
Finally, we note that
XA
Æ1
lim (lnX
A) =
XA
Æ1
lim (ln (1–X
B)) = – X
B
Substituting these various limiting values
into Eq. (1-132) yields:
XA
Æ1
lim (dX
l
A
/dT– dX
s
A
/dT)
= Dh
0
f(A)
/R(T
f(A))
2
(1-133)
If the limiting slope of the liquidus,
XA
Æ1
lim (dX
l
A
/dT), is known, then the limiting
slope of the solidus can be calculated, or
vice versa, as long as the enthalpy of fusion
is known.
For the LiF–NaF system in Fig. 1-18,
the broken line is the limiting liquidus
slope at X
LiF=1 calculated from Eq. (1-
133) under the assumption that there is no
solid solubility (that is, that dX
s
A
/dT= 0).
Agreement with the measured limiting liq-
uidus slope is very good, thereby showing
that the solid solubility of NaF in LiF is not
large.
In the general case, the solute B may dis-
solve to form more than one “particle”. For
example, in dilute solutions of Na
2SO
4in
MgSO
4, each mole of Na
2SO
4yields two
moles of Na
+
ions which mix randomly
with the Mg
2+
ions on the cationic sublat-
tice. Hence, the real mole fraction of sol-
vent, X
A, is (1–nX
B) where nis the number
of moles of foreign “particles” contributed
by one mole of solute. In the present exam-
ple,
n=2.
Eq. (1-133) now becomes:
XA
Æ1
lim (dX
l
A
/dT– dX
s
A
/dT)
= Dh
0
f(A)
/nR(T
f(A))
2
(1-134)
The broken line in Fig. 1-44 is the limiting
liquidus slope calculated from Eq. (1-134)
under the assumption of negligible solid
solubility.
It can be shown (Blander, 1964) that Eq.
(1-134) applies very generally with the fac-
tor
nas defined above. For example, add-
ing LiF to NaF introduces only one foreign
Figure 1-44.Phase dia-
gram of the MgSO
4–
Na
2SO
4system calculated
for an ideal ionic liquid so-
lution. Broken line is the
theoretical limiting liquidus
slope calculated for negli-
gible solid solubility taking
into account the ionic nature
of the liquid. Agreement
with the measured diagram
(Ginsberg, 1909) is good.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

79. particle Li
+
. The F
–
ion is not “foreign”.
Hence,
n=1. For additions of Na
2SO
4to
MgSO
4, n= 2 since two moles of Na
+
ions
are supplied per mole of Na
2SO
4. For
CaCl
2dissolved in water, n= 3, and so on.
For C dissolving interstitially in solid Fe,
n=1. The fact that the solution is intersti-
tial has no influence on the validity of Eq.
(1-134). Eq. (1-134) is thus very general
and very useful. It is independent of the so-
lution model and of the excess properties,
which become zero at infinite dilution.
An equation identical to Eq. (1-134) but
with the enthalpy of transition, Dh
0
tr
, re-
placing the enthalpy of fusion, relates the
limiting phase boundary slopes at a trans-
formation temperature of a component.
1.10.4 Short-Range Ordering
The basic premise of the regular solution
model (Sec. 1.5.7) is that random mixing
occurs even when g
E
is not zero. To ac-
count for non-random mixing, the regular
solution model has been extended though
the quasichemical model for short-range
orderingdeveloped by Guggenheim (1935)
and Fowler and Guggenheim (1939) and
modified by Pelton and Blander (1984,
1986) and Blander and Pelton (1987). The
model is outlined below. For a more com-
plete development, see the last two papers
cited above, Degterov and Pelton (1996),
Pelton et al. (2000) and Pelton and Char-
trand (2000).
For a binary system, consider the for-
mation of two nearest-neighbor 1–2 pairs
from a 1–1 and a 2–2 pair:
(1–1) + (2–2) = 2(1–2) (1-135)
Let the molar Gibbs energy change for
this reaction be (
w–hT). Let the nearest-
neighbor coordination numbers of 1 and 2
atoms or molecules be Z
1and Z
2. The total
number of bonds emanating from an iatom
or molecule is Z
iX
i. Hence, mass balance
equations can be written as
Z
1X
1= 2n
11+ n
12
Z
2X
2= 2n
22+ n
12 (1-136)
where n
ijis the number of i–jbonds in one
mole of solution. “Coordination equivalent
fractions” may be defined as:
Y
1=1– Y
2= Z
1X
1/(Z
1X
1+ Z
2X
2) (1-137)
where the total number of pairs in one mole
of solution is (Z
1X
1+Z
2X
2)/2. Letting X
ij
be the fraction of i–jpairs in solution, Eq.
(1-136) may be written as:
2Y
1= 2X
11+ X
12
2Y
2= 2X
22+ X
12 (1-138)
The molar enthalpy and excess entropy of
mixing are assumed to be directly related
to the number of 1–2 pairs:
Dh
m– Ts
E (non-config)
= (Z
1X
1+ Z
2X
2)X
12(w–hT)/4 (1-139)
An approximate expression for the config-
urational entropy of mixing is given by a
one-dimensional Ising model:
The equilibrium distribution is calculated
by minimizing Dg
mwith respect to X
12at
constant composition. This results in a
“quasichemical” equilibrium constant for
the reaction, Eq. (1-135):
When (
w–hT) = 0, the solution of Eqs. (1-
138) and (1-141) gives a random distribu-
X
XX
T
RT
12
2
11 22
4= (1-141)exp
()−−⎛
⎝
⎜
⎞
⎠
⎟
wh
DsRXXXX
R
ZX ZX
XXYXXY
XXYY
m
config=
(1-140
)
−+
−+
×+
+
(ln ln)
()
[ ln( / ) ln( / )
ln( / )]
112 2
11 2 2
11 11 1
2
22 22 2
2
12 12 1 2
2
2
68 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
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80. 1.10 Solution Models 69
tion with X
11=Y
1
2, X
22=Y
2
2and 2Y
1Y
2, and
Eq. (1-140) reduces to the ideal Raoultian
entropy of mixing. When (
w–hT) be-
comes very negative, 1–2 pairs predomi-
nate. A plot of D h
mor s
E (non-config)
versus
composition then becomes V-shaped and a
plot of D s
m
configbecomes m-shaped, with
minima at Y
1=Y
2=1/2, which is the com-
position of maximum ordering, as illus-
trated in Fig. 1-45. When (
w–hT) is quite
negative, the plot of g
E
also has a negative
V-shape.
For Fe–S liquid solutions, the activity
coefficients of sulfur as measured by sev-
eral authors are plotted in Fig. 1-46. It is
clear in this case that the model should be
applied with Z
Fe=Z
S. The curves shown in
Fig. 1-46 were calculated from the quasi-
chemical model with (
w–hT) expanded as
the following optimized (Kongoli et al.,
1998) polynomial:
(1-142)
(
w–hT) = – (70 017 + 9T) – 74 042Y
S
– (798 – 15T)Y
S
3+ 40 791Y
S
7J/mol
–1
Far fewer parameters are required than if a
polynomial expansion of g
E
(as in Sec.
1.6.2) were used. Furthermore, and more
importantly, the model permits successful
predictions of the properties of multicom-
ponent systems as illustrated in Fig. 1-47,
where measured sulfur activities in quater-
nary liquid Fe–Ni–Cu–S solutions are
compared with activities calculated (Kon-
goli et al., 1998) solely from the optimized
model parameters for the Fe–S, Ni–S
and Cu–S binary systems. A pair exchange
reaction like Eq. (1-135) was assumed
for each M–S pair (M = Fe, Ni, Cu), and
an optimized polynomial expansion of
(
w
MS–h
MST) as a function of Y
S, similar to
Eq. (1-142), was obtained for each binary.
Three equilibrium constant equations like
Eq. (1-141) were written, and it was as-
sumed that (
w
MS–h
MST) in the quaternary
system was constant at constant Y
S. No ad-
justable ternary or quaternary parameters
were required to obtain the agreement
shown in Fig. (1-47), although the model
permits the inclusion of such terms if re-
quired.
Silicate slags are known to exhibit such
short-range ordering. In the CaO–SiO
2
system, Dh
mhas a strong negative V-shape,
as in Fig. 1-45, but with the minimum at
X
SiO
2
= 1/3 which is the composition corre-
sponding to Ca
2SiO
4. That is, the ordering
is associated with the formation of ortho-
silicate anions SiO
4
4–. In the phase diagram,
Fig. 1-14, the CaO-liquidus can be seen to
descend sharply near the composition
X
SiO
2
= 1/3. The quasichemical model has
been extended by Pelton and Blander
(1984) to treat silicate slags. The diagram
Figure 1-45.Molar enthalpy and entropy of mixing
curves for a system AB calculated at 1000°C with
Z
1=Z
2=2 from the quasichemical model for short-
range ordering with (
ω–ηT) = 0, – 21, – 42, and
– 84 kJ.www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

81. 70 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-46.Activity coefficient of sulfur in liquid Fe – S solutions calculated from optimized quasichemical
model parameters and comparison with experimental data (Kongoli et al., 1998).
Figure 1-47.Equilibrium partial pressure of sulfur at 1200
o
C over Fe – Ni – Cu – S mattes predicted by the
quasichemical model from binary data (Kongoli et al., 1998) and comparison with experimental data
(Bale and Toguri, 1996).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

82. 1.10 Solution Models 71
shown in Fig. 1-14 is thermodynamically
calculated (Wu, 1990).
Many liquid alloy solutions exhibit
short-range ordering. The ordering is
strongest when one component is relatively
electropositive (on the left side of the peri-
odic table) and the other is relatively elec-
tronegative. Liquid alloys such as Alk–Au
(Hensel, 1979), Alk–Pb (Saboungi et al.,
1985) and Alk–Bi (Petric et al., 1988a),
where Alk = (Na, K, Rb, Cs), exhibit curves
of Dh
mand Ds
msimilar to those in Fig.
1-45 with one composition of maximum
ordering. For example, in the Au–Cs sys-
tem the minima occur near the composition
AuCs; in Mg–Bi alloys the minima occur
near the Mg
3Bi
2composition, while in
K–Pb alloys the maximum ordering is at
K
4Pb.
It has also been observed that certain liq-
uid alloys exhibit more than one composi-
tion of ordering. For example, in K–Te al-
loys, the “excess stability function”, which
is the second derivative of Dg
m, exhibits
peaks near the compositions KTe
8, KTe
and K
2Te (Petric et al., 1988b) thereby
providing evidence of ordering centred on
these compositions. The liquid might be con-
sidered as consisting of a series of mutually
soluble “liquid intermetallic compounds”.
When (
w–hT) is expanded as a polyno-
mial as in Eq. (1-142), the quasichemical
model and the polynomial model of Sec.
1.6.2 become identical as (
w–hT) ap-
proaches zero. That is, the polynomial
model is a limiting case of the quasichemi-
cal model when the assumption of ideal
configurational entropy is made.
When (
w–hT) is positive, (1–1) and
(2–2) pairs predominate. The quasichemi-
cal model can thus also treat such cluster-
ing, which accompanies positive devia-
tions from ideality.
Recent work (Pelton et al., 2000; Pelton
and Chartrand, 2000) has rendered the
model more flexible by permitting the Z
ito
vary with composition and by expanding
the (
w–hT) as polynomials in the bond
fractions X
ijrather than the overall compo-
nent fractions. A merger of the quasichem-
ical and sublattice models has also been
completed (Chartrand and Pelton, 2000),
permitting nearest-neighbor and second-
nearest neighbor short-range-ordering to
be treated simultaneously in molten salt so-
lutions.
1.10.5 Long-Range Ordering
In solid solutions, long-range ordering
can occur as well as short-range ordering.
In Fig. 1-15 for the Ag–Mg system, a
transformation from an a¢ to an aphase is
shown occurring at approximately 665 K
at the composition Ag
3Mg. This is an
order–disordertransformation. Below the
transformation temperature, long-range or-
dering(superlattice formation) is observed.
An order parametermay be defined which
decreases to zero at the transformation
temperature. This type of phase transfor-
mation is not a first-order transformation
like those considered so far in this chapter.
Unlike first-order transformations which
involve a change of state (solid, liquid, gas)
and also involve diffusion over distances
large compared with atomic dimensions,
order–disorder transformations, at least at
the stoichiometric composition (Ag
3Mg in
this example), occur by atomic rearrange-
ment over distances of the order of atomic
dimensions. The slope of the curve of
Gibbs energy versus Tis not discontinuous
at the transformation temperature. Order-
ing and order–disorder transformations are
discussed in Chapter 8 (Inden, 2001).
A type of order–disorder transformation
of importance in ferrous metallurgy is the
magnetic transformation. Below its Curie
temperature of 769 °C, Fe is ferromagnetic.www.iran-mavad.com
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83. Above this temperature it is not. The trans-
formation involves a change in ordering of
the atomic spins and is not first order. Ad-
ditions of alloying elements will change
the temperature of transformation. Mag-
netic transformations are treated in Chapter
4 (Binder, 2001). See also Miodownik
(1982) and Inden (1982).
1.11 Calculation of Ternary Phase
Diagrams From Binary Data
Among 70 metallic elements 70!/3!67!
= 54 740 ternary systems and 916 895 qua-
ternary systems are formed. In view of the
amount of work involved in measuring
even one isothermal section of a relatively
simple ternary phase diagram, it is very im-
portant to have a means of estimating ter-
nary and higher-order phase diagrams.
The most fruitful approach to such pre-
dictions is via thermodynamic methods. In
recent years, great advances have been
made in this area by the international Cal-
phad group. Many key papers have been
published in the Calphad Journal.
As a first step in the thermodynamic ap-
proach, we critically analyze the experi-
mental phase diagrams and thermodynamic
data for the three binary subsystems of the
ternary system in order to obtain a set of
mathematical expressions for the Gibbs
energies of the binary phases, as was dis-
cussed in Sec. 1.6. Next, interpolation pro-
cedures based on solution models are used
to estimate the Gibbs energies of the ter-
nary phases from the Gibbs energies of the
binary phases. Finally, the ternary phase di-
agram is calculated by computer from
these estimated ternary Gibbs energies by
means of common tangent plane or total
Gibbs energy minimization algorithms.
As an example of such an estimation of
a ternary phase diagram, the experimental
(Ivanov, 1953) and estimated (Lin et al.,
1979) liquidus projections of the KCl–
MgCl
2–CaCl
2system are shown in Fig.
1-48. The estimated phase diagram was
calculated from the thermodynamic prop-
erties of the three binary subsystems with
the Gibbs energy of the ternary liquid ap-
proximated via the equation suggested by
72 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-48.Projection of the liquidus surface of
the KCl – MgCl
2– CaCl
2system.
a) Calculated from optimized binary thermodynamic
parameters (Lin et al., 1979).
b) As reported by Ivanov (1953).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

84. 1.11 Calculation of Ternary Phase Diagrams From Binary Data 73
Kohler (1960):
g
E
= (1 – X
A)
2
g
E
B/C
+ (1 – X
B)
2
g
E
C/A
+ (1 – X
C)
2
g
E
A/B
(1-143)
In this equation, g
E
is the excess molar
Gibbs energy at a composition point in the
ternary liquid phase and g
E
B/C
, g
E
C/A
and g
E
A/B
are the excess Gibbs energies in the three
binary systems at the same ratios X
B/X
C,
X
C/X
Aand X
A/X
Bas at the ternary point. If
the ternary liquid phase as well as the three
binary liquid phases are all regular solu-
tions, then Eq. (1-143) is exact. In the
general case, a physical interpretation of
Eq. (1-143) is that the contribution to g
E
from, say, pair interactions between A and
B particles is constant at a constant ratio
X
A/X
Bapart from the dilutive effect of the
C particles, which is accounted for by the
term (1–X
C)
2
taken from regular solution
theory.
Ternary phase diagrams estimated in this
way are quite acceptable for many pur-
poses. The agreement between the experi-
mental and calculated diagrams can be
greatly improved by the inclusion of one
or two “ternary terms” with adjustable
coefficients in the interpolation equations
for g
E
. For example, the ternary term
aX
KClX
MgCl
2
X
CaCl
2
, which is zero in all
three binaries, could be added to Eq. (1-
143) and the value of the parameter a
which gives the “best” fit to the measured
ternary liquidus could be determined. This,
of course, requires that ternary measure-
ments be made, but only a very few (even
one or two in this example) experimental
liquidus points will usually suffice rather
than the larger number of measurements
required for a fully experimental determi-
nation. In this way, the coupling of the
thermodynamic approach with a few well
chosen experimental measurements holds
promise of greatly reducing the experimen-
tal effort involved in determining multi-
component phase diagrams.
Reviews of various interpolation proce-
dures and computer techniques for estimat-
ing and calculating ternary and higher-or-
der phase diagrams are given by Ansara
(1979), Spencer and Barin (1979) and Pel-
ton (1997).
Other equations, similar to the Kohler
Eq. (1-143) in that they are based on exten-
sion of regular solution theory, are used to
estimate the thermodynamic properties of
ternary solutions from the properties of the
binary subsystems. For a discussion and
references, see Hillert (1980). However,
for structurally more complex solutions
involving more than one sublattice or with
significant structural ordering, other esti-
mation techniques must be used. For a re-
view, see Pelton (1997).
An example, the calculation of the
phase diagram of the NaCl–KCl–NaF–KF
system in Fig. 1-31, has already been pre-
sented in Sec. 1.10.1.2.
The quasichemical model for systems
with short-range ordering was discussed
for the case of binary systems in Sec.
1.10.4. The model has been extended to
permit the estimation of ternary and multi-
component phase diagrams (Pelton and
Blander, 1986; Blander and Pelton, 1987;
Pelton and Chartrand, 2000). Very good re-
sults have been obtained in the case of sili-
cate systems. The liquidus surface of the
SiO
2–MgO–MnO system, estimated from
optimized binary data with the quasichem-
ical model for the liquid and under the
assumption of ideal mixing for the solid
MgSiO
3–MnSiO
3and Mg
2SiO
4–Mn
2SiO
4
solutions, is shown in Fig. 1-49. Agree-
ment with the measured phase diagram
(Glasser and Osborn, 1960) is within ex-
perimental error limits.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

85. 1.12 Minimization of Gibbs
Energy
Throughout this chapter it has been
shown that phase equilibria are calculated
by Gibbs energy minimization. Computer
software has been developed in recent
years to perform such calculations in sys-
tems of any number of components, phases
and species.
Consider a system in which several stoi-
chiometric solid or liquid compounds A, B,
C, … could be present at equilibrium along
with a number of gaseous, liquid or solid
solution phases a, b, g, …. The total Gibbs
energy of the system may be written as:
G= (n
Ag
0
A
+ n
Bg
0
B
+ …)
+ (n
ag
a+ n
bg
b+ …) (1-144)
where n
A, n
B, etc. are the number of moles
of the pure solid or liquidus; g
0
A
, g
0
B
, etc.
are the molar Gibbs energies of the pure
solids or liquids (which are functions of T
and P); n
a, n
b, etc. are the total number of
moles of the solution phases; g
a, g
b, etc.
are the molar Gibbs energies of the solution
phases (which are function of T, Pand
composition). For a given set of constraints
74 1 Thermodynamics and Phase Diagrams of Materials
Figure 1-49.Projection of liquidus surface of the SiO
2–MgO–MnO system calculated from optimized binary
parameters with the quasichemical model for the liquid phase.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

86. 1.12 Minimization of Gibbs Energy 75
(such as fixed T, Pand overall composi-
tion), the free energy minimization algo-
rithms find the set of mole numbers n
A, n
B,
etc., n
a, n
b, etc. (some may be zero) as well
as the compositions of all solution phases
which globally minimize G. This is the
equilibrium phase assemblage. Other con-
straints such as constant volume or a fixed
chemical potential (such as constant p
O
2
)
may be applied.
A discussion of the strategies of such al-
gorithms is beyond the scope of the present
chapter. One of the best known general
Gibbs energy minimization programs is
Solgasmix written by Eriksson (1975) and
constantly updated.
When coupled to a large thermodynamic
database, general Gibbs energy minimiza-
tion programs provide a powerful tool for
the calculation of phase equilibria. Several
such expert database systems have been
developed. They have been reviewed by
Bale and Eriksson (1990).
An example of a calculation performed
by the F*A*C*T (Facility for the Analy-
sis of Chemical Thermodynamics) expert
system, which the author has helped to de-
velop, is shown in Table 1-2. The program
has been asked to calculate the equilibrium
state when 1 mol of SiI
4is held at 1400 K
in a volume of 10
4
l. The thermodynamic
properties of the possible product species
have been automatically retrieved from
the database and the total Gibbs energy
has been minimized by the Solgasmix
algorithm. At equilibrium there will be
2.9254 mol of gas of the composition
shown and 0.11182 mol of solid Si will
precipitate. The total pressure will be
0.0336 bar.
Although the calculation was performed
by minimization of the total Gibbs energy,
substitution of the results into the equilib-
rium constants of Eqs. (1-10) to (1-12) will
show that these equilibrium constants are
satisfied.
Another example is shown in Table 1-3
(Pelton et al., 1990). Here the program has
been asked to calculate the equilibrium
Table 1-2.Calculation of equilibrium state when
1 mole SiI
4is held at 1400 K in a volume of 10
4
l.
Calculations performed by minimization of the total
Gibbs energy.
SiI
4=
2.9254 ( 0.67156 I
+ 0.28415 SiI
2
+ 0.24835E – 01 I
2
+ 0.19446E – 01 SiI
4
+ 0.59083E – 05 SiI
+ 0.23039E – 07 Si
+ 0.15226E – 10 Si
2
+ 0.21462EE – 11 Si
3)
(1400.0,0.336E – 01,G)
+ 0.11182 Si
(1400.0,0.336E – 01,S1, 1.0000)
Table 1-3.Calculation of equilibrium state when re-
actants shown (masses in g) are held at 1873 K at a pressure of 1 atm. Calculations performed by mini-
mization of the total Gibbs energy.
100. Fe + 0.08 O + 0.4 Fe + 0.4 Mn + 0.3 Si + 0.08 Ar =
0.30793 litre ( 99.943 vol% Ar
+ 0.24987E – 01 vol% Mn
+ 0.24069E – 01 vol% SiO
+ 0.82057E – 02 vol% Fe
+ 0.79044E – 07 vol% O
+ 0.60192E – 08 vol% Si
+ 0.11200E – 08 vol% O
2
+ 0.35385E – 15 vol% Si
2)
(1873.0, 1.00 ,G)
+ 0.18501 gram ( 49.948 wt.% SiO
2
+ 42.104 wt.% MnO
+ 7.9478 wt.% FeO)
(1873.0, 1.00 ,SOLN 2)
+ 100.99 gram ( 99.400 wt.% Fe
+ 0.33630 wt.% Mu
+ 0.25426 wt.% Si
+ 0.98375E – 02 wt.% O
2)
(1873.0, 1.00 ,SOLN 3)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

87. state when 100 g Fe, 0.08 g oxygen, 0.4 g
Fe, 0.4 g Mn, 0.3 g Si and 0.08 g Ar are
brought together at 1873 K at a total pres-
sure of 1 bar. The database contains data
for a large number of solution phases as
well as for pure compounds. These data
have been automatically retrieved and the
total Gibbs energy has been minimized. At
equilibrium there are 0.30793 l of a gas
phase, 0.18501 g of a molten slag, and
100.99 g of a molten steel of the composi-
tions shown.
The Gibbs energies of the solution
phases are represented as functions of com-
position by various solution models (Sec.
1.10). As discussed in Sec. 1.11, these
models can be used to predict the thermo-
dynamic properties of N-component solu-
tions from evaluated parameters for binary
(and possibly ternary) subsystems stored in
the database. For example, in the calcula-
tion in Table 1-3, the Gibbs energy of the
molten slag phase was estimated by the
quasichemical model from optimized pa-
rameters for the binary oxide solutions.
1.12.1 Phase Diagram Calculation
Gibbs energy minimization is used to
calculate general phase diagram sections
thermodynamically using the zero phase
fraction line concept (Sec. 1.9.1.1), with
data retrieved from databases of model co-
efficients. For example, to calculate the di-
agram of Fig. 1-30, the program first scans
the four edges of the diagram to find the
ends of the ZPF lines. Each line is then fol-
lowed from beginning to end, using Gibbs
energy minimization to determine the point
at which a phase is just on the verge of be-
ing present. When ZPF lines for all phases
have been drawn, then the diagram is com-
plete. Because, as shown in Sec. 1.9, all
true phase diagram sections obey the same
geometrical rules, one algorithm suffices to
calculate all types of phase diagrams with
any properly chosen variables as axes or
constants.
1.13 Bibliography
1.13.1 Phase Diagram Compilations
The classic compilation in the field of bi-
nary alloy phase diagrams is that of Hansen
(1958). This work was continued by Elliott
(1965) and Shunk (1969). These compila-
tions contain critical commentaries. A non-
critical compilation of binary alloy phase
diagrams is supplied in looseleaf form with
a continual up-dating service by W.G. Mof-
fatt of the General Electric Co., Schenec-
tady, N.Y. An extensive non-critical compi-
lation of binary and ternary phase diagrams
of metallic systems has been edited by
Ageev (1959 – 1978). An index to all com-
pilations of binary alloy phase diagrams up
to 1979 was prepared by Moffatt (1979). A
critical compilation of binary phase dia-
grams involving Fe has been published by
Kubaschewski (1982). Ternary alloy phase
diagrams were compiled by Ageev (1959–
1978).
From 1979 to the early 1990s, the Amer-
ican Society for Metals undertook a project
to evaluate critically all binary and ternary
alloy phase diagrams. All available litera-
ture on phase equilibria, crystal structures,
and often thermodynamic properties were
critically evaluated in detail by interna-
tional experts. Many evaluations have ap-
peared in the Journal of Phase Equilibria
(formerly Bulletin of Alloy Phase Dia-
grams), (ASM Int’l., Materials Park, OH),
which continues to publish phase diagram
evaluations. Condensed critical evalua-
tions of 4700 binary alloy phase diagrams
have been published in three volumes
(Massalski et al., 1990). The ternary phase
76 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

88. 1.13 Bibliography 77
diagrams of 7380 alloy systems have also
been published in a 10-volume compilation
(Villars et al., 1995). Both binary and ter-
nary compilations are available from ASM
on CD-ROM. Many of the evaluations
have also been published by ASM as
monographs on phase diagrams involving a
particular metal as a component.
Each year, MSI Services (http://www.
msiwp.com) publishes The Red Book,
which contains abstracts on alloy phase
diagrams from all sources, notably from
the extensive Russian literature. MSI also
provides a CD-ROM with extensive alloy
phase diagram compilations and reports.
Phase diagrams for over 9000 binary, ter-
nary and multicomponent ceramic systems
(including oxides, halides, carbonates, sul-
fates, etc.) have been compiled in the 12-
volume series, Phase Diagrams for Ceram-
ists (1964–96, Am. Ceramic Soc., Colum-
bus, OH). Earlier volumes were non-criti-
cal compilations. However, recent volumes
have included critical commentaries.
Phase diagrams of anhydrous salt sys-
tems have been compiled by Voskresen-
skaya (1970) and Robertson (1966).
An extensive bibliography of binary
and multicomponent phase diagrams of all
types of systems (metallic, ceramic, aque-
ous, organic, etc.) has been compiled by
Wisniak (1981).
1.13.2 Thermodynamic Compilations
Several extensive compilations of ther-
modynamic data of pure substances of
interest in materials science are available.
These include the JANAF Tables (Chase
et al., 1985) and the compilations of Barin
et al. (1977), Barin (1989), Robie et al.
(1978) and Mills (1974), as well as the
series of compilations of the National Insti-
tute of Standards and Technology (Wash-
ington, D.C.).
Compilations of properties of solutions
(activities, enthalpies of mixing, etc.) are
much more difficult to find. Hultgren et al.
(1973) present the properties of a number
of binary alloy solutions. An extensive bib-
liography of solution properties of all types
of solutions was prepared by Wisniak and
Tamir (1978).
Thermodynamic/phase diagram optimi-
zation as discussed in Sec. 1.6.1 has been
carried out for a large number of alloy, ce-
ramic and other systems. Many of these
evaluations have been published in the
international Calphad Journal, published
since 1977 by Pergamon Press. Several of
the evaluations in the Journal of Phase
Equilibriadiscussed above include ther-
modynamic/phase diagram optimizations,
as do a number of the evaluations in Vol. 7
of Phase Diagrams for Ceramists.
Extensive computer databases of the
thermodynamic properties of compounds
and solutions (stored as coefficients of
model equations) are available. These in-
clude F*A*C*T (http://www.crct.poly-
mtl.ca), Thermocalc (http://www.met.
kth.se), ChemSage (http://gttserv.lth.rwth-
aachen.de), MTS-NPL (http://www.npl.
co.uk), Thermodata (http://www.grenet.fr),
HSC (http://www.outokumpu.fi), and
MALT2 (http://www. kagaku.com). Gibbs
energy minimization software permits the
calculation of complex equilibria from the
stored data as discussed in Sec. 1.12 as
well as the thermodynamic calculation of
phase diagram sections. A listing of these
and other available databases is maintained
at http://www.crct. polymtl.ca .
A bibliographic database known as
Thermdoc, on thermodynamic properties
and phase diagrams of systems of interest
to materials scientists, with updates, is
available through Thermodata (http://
www.grenet.fr).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

89. 1.13.3 General Reading
The theory, measurement and applica-
tions of phase diagrams are discussed in a
great many texts. Only a few can be listed
here. A recent text by Hillert (1998) pro-
vides a complete thermodynamic treatment
of phase equilibria as well as solution mod-
eling and thermodynamic/phase diagram
optimization.
A classical discussion of phase diagrams
in metallurgy was given by Rhines (1956).
Prince (1966) presents a detailed treatment
of the geometry of multicomponent phase
diagrams. A series of five volumes edited
by Alper (1970–1978) discusses many as-
pects of phase diagrams in materials sci-
ence. Bergeron and Risbud (1984) give an
introduction of phase diagrams, with par-
ticular attention to applications in ceramic
systems, see also Findlay (1951), Ricci
(1964) and West (1965).
In the Calphad Journal and in the Jour -
nal of Phase Equilibriaare to be found
many articles on the relationships between
thermodynamics and phase diagrams.
It has been beyond the scope of the
present chapter to discuss experimental
techniques of measuring thermodynamic
properties and phase diagrams. For the
measurement of thermodynamic proper-
ties, including properties of solutions, the
reader is referred to Kubaschewski and
Alcock (1979). For techniques of measur-
ing phase diagrams, see Pelton (1996),
Raynor (1970), MacChesney and Rosen-
berg (1970), Buckley (1970) and Hume-
Rothery et al. (1952).
1.14 References
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Neorg. Khim. 2,669.
Alper, A. M. (Ed.) (1970–1978), Phase Diagrams –
Materials Science and Technology, Vols. 1–5.
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Bale, C. W., Pelton, A. D. (1983), Metall. Trans.
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80 1 Thermodynamics and Phase Diagrams of Materialswww.iran-mavad.com
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92. 10 High Pressure Phase Transformations
Martin Kunz
ETH Zürich, Labor für Kristallographie, Zürich, Switzerland
List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 657
10.1Introduction................................. 659
10.2Pressure-Driven Phase Transitions.................... 660
10.2.1 Framework flexion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
10.2.2 Increase in coordination number . . . . . . . . . . . . . . . . . . . . . . . 663
10.2.3 Pressure-induced ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 664
10.3Generating High Pressure......................... 666
10.3.1 Dynamic pressure generation . . . . . . . . . . . . . . . . . . . . . . . . 667
10.3.2 Static pressure devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
10.3.2.1 Large-volume presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
10.3.2.2 Diamond anvil cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671
10.4Probing Phase Transformations in Materials at High Pressure..... 673
10.4.1 Volumetric techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
10.4.2 Spectroscopic techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 674
10.4.2.1 Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
10.4.2.2 Raman and infrared spectroscopy . . . . . . . . . . . . . . . . . . . . . . 674
10.4.2.3 Mössbauer spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
10.4.2.3 X-Ray absorption spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 676
10.4.2.5 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
10.4.3 Ultrasonic sound velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 677
10.4.4 Diffraction techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678
10.4.4.1 X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678
10.4.4.2 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
10.5Examples.................................. 679
10.5.1 Zincblende-type semiconductors . . . . . . . . . . . . . . . . . . . . . . 679
10.5.1.1 Si and Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
10.5.1.2 GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681
10.5.1.3 InSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
10.5.2 Materials in the B-C-N system . . . . . . . . . . . . . . . . . . . . . . . . 683
10.5.2.1 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
10.5.2.2 B–N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
10.5.2.3 C–N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
10.5.2.4 B–C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
10.5.2.5 B–C–N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
10.5.3 H
2O ..................................... 687
10.5.3.1 Ice Ih, XI and Ic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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93. 10.5.3.2 Ice II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
10.5.3.3 Ice III and Ice IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
10.5.3.4 Ice V, Ice IV and Ice XII . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
10.5.3.5 Ice VI, Ice VII, Ice VIII, Ice X . . . . . . . . . . . . . . . . . . . . . . . 690
10.5.3.6 Amorphous ice at high pressure . . . . . . . . . . . . . . . . . . . . . . . 691
10.6Acknowledgment .............................. 692
10.7References .................................. 692
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94. List of Symbols and Abbreviations 657
List of Symbols and Abbreviations
a,b,c lattice parameters
E
0 internal energy at zero pressure in shock experiment
E
H Hugoniot internal energy in a shock experiment
G Gibbs energy
H enthalpy
l wavelength
m chemical potential
n number of moles
P pressure
P
0 ambient pressure
P
H Hugoniot pressure
Q heat
r
0 density at zero pressure
S entropy
T temperature
T
c critical temperature in superconductors
T
0 temperature at zero pressure
T
H Hugoniot temperature
U internal energy
U
P sample velocity
U
s shock velocity
V molar volume
V
0 molar volume at zero pressure
V
H molar volume in the shocked Hugoniot state
DV
dis difference in molar volume due to an order/disorder process
V
disorderedmolar volume of the disordered state
V
ordered molar volume of the ordered state
W work
ADX angle dispersive X-ray diffraction
AX compound semiconductors of 1:1 stoichiometry
bcc body centered cubic
ccp cubic closest packing (= fcc)
ct carat
DAC diamond anvil cell
dhcp hexagonal closest packing with doubled c-axis
EDX energy dispersive X-ray diffraction
EXAFS extended X-ray absorption fine struture
fcc face centered cubic (=ccp)
hcp hexagonal closest packing
HDA high density amorphous phase of H
2O
HEL Hugoniot elastic limit
HP high pressurewww.iran-mavad.com
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95. HT high temperature
IR infrared
LDA low-density amorphous phase of H
2O
NMR nuclear magnetic resonance
RT room temperature
XAS X-ray absorption spectroscopy
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96. 10.1 Introduction 659
10.1 Introduction
The understanding of condensed matter
requires a knowledge of the relationship
between temperature, pressure and chemi-
cal environment on the one hand and vol-
ume, bonding and electronic and magnetic
structure on the other. In gases, pressure,
temperature and volume largely follow a
simple and universal relationship, but the
situation in condensed matter is much more
complicated. This is mainly due to inter-
atomic interactions, which are much
stronger in condensed material than in
gases and cause structural, electronic and
magnetic correlations. Pelton (2001, Chap. 1
of this book) explores the interdependence
between chemical composition and tem-
perature in great detail. In this chapter, we
will mainly focus on the effects of pressure
on the state of solid material.
The exploration of the physics and
chemistry of material under high pressure
is of a remarkably inter-disciplinary inter-
est. This is reflected in the variety of scien-
tific problems that are the focus of modern
high-pressure research. The understanding
of the interdependence between structural
distortions and electronic and magnetic
properties is a typical example of a physi-
cal problem that benefits from experiments
performed at high pressure. Another topic
within physics that relies on high-pressure
experiments is the calibration and refine-
ment of theoretical models describing the
interaction of atoms. High-pressure experi-
ments on very simple covalent systems
such as solid hydrogen or helium provide
valuable data, which help to formulate fun-
damental concepts on the nature of matter.
A fascinating problem in condensed-matter
chemistry is the structural change of a solid
(or liquid) and its phase transitions as a re-
action on an external pressure. Its under-
standing offers chemists vital insights into
the thermodynamics of solid or liquid
material (see also the Chapter by Binder,
2001). In a more practical way, materials
scientists are interested in adding pressure
to the variables temperature and chemical
composition in order to stabilize new mate-
rials with technically interesting proper-
ties. A prominent example of a class of ma-
terials whose synthesis and exploration de-
pend on the application of high pressure is
the family of novel abrasives and super-
hard materials. In recent years, high-pres-
sure experiments are even used in the life
sciences. A popular key issue in this area
is the question on the origin of life (e.g.,
Pedersen, 1997). A more applied interest
of biology in high pressure is the search
for commercially viable alternative ways
of food sterilization (e.g., Ondrey and Ka-
miya, 2000; Thakur and Nelson, 1998;
Mermelstein, 1999).
Forcing a given set of atoms into a small
volume by applying high pressure in-
creases the interatomic interaction. This in
turn leads to structural phase transitions
and changes in physical properties. The
driving forces for high pressure-induced
phase transitions are obviously linked to
the need to optimize the volume occupied
by the atoms for a given pressure. In Sec.
10.2, we will discuss the most important
mechanisms, which are the immediate
cause of pressure-driven phase transitions.
Extracting information on solid or liquid
material at high pressure (“high” pressure
in this chapter refers to pressures in the
range 0.1 –100 GPa) is an experimental
challenge and was first tackled by Percy
William Bridgman in the first half of this
century (e.g., Bridgman, 1946). He was
awarded the Nobel Prize for physics in
1946 for his ground-breaking achieve-
ments in this important field. Since the
time of Bridgman the technology necessary
to perform experiments at high pressurewww.iran-mavad.com
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97. has experienced extensive development.
Section 10.3 will briefly summarize the
current state of the art of high-pressure ex-
perimentation. Although third-generation
synchrotron sources produced a minor rev-
olution in high-pressure crystallography,
X-ray diffraction is by no means the only
way of investigating material when sub-
jected to pressure. In Sec. 10.4, we will re-
view the most common probes used for
physical and chemical investigations at
high pressure. Section 10.5 treats in detail
the high-pressure phase transitions for a se-
lected sample of materials, which are of
special interest within materials science.
10.2 Pressure-Driven Phase
Transitions
The physical quantities that define the
thermodynamic state of a system can be
differentiated into intensive quantities (i.e.,
temperature T , pressure P, chemical poten-
tial µ) and extensive quantities (i.e., en-
tropy S, volume V, number of moles n).
Conjugate quantities are pairs of an inten-
sive and an extensive variable (i.e., T,Sor
P,V). Their products have the dimensions
of energy or a volume-normalized energy,
which can be used to describe the relative
stability of a given system. The thermody-
namic stability is determined by the mini-
mum of the internal energy U, which is de-
fined as a sum of a heat term Q and a work
term W,
U= Q+ W (10-1)
where Qis a function of temperature and
entropy and Wis a function of pressure and
volume. On an atomistic level, Qcan be
viewed as the vibrational energy of the at-
oms oscillating around their equilibrium
position. Wcan be visualized as the sum of
the potential energies from the inter atomic
interactions between the ensemble of con-
stituting atoms.
The absolute value of the internal energy
cannot be measured. However, the differ-
ence between two states is independent of
the path and mechanism of the change of
the system (Hess’s law). This makes the
difference in the internal energy dU a very
important quantity in comparing a system
at two different states:
dU= TdS– PdV (10-2)
Entropy Sand volume Vare two indepen-
dent variables that are very difficult to con-
trol in an experiment. However, two suc-
cessive Legendre transformations of U
transform the internal energy first into the
enthalpy H
H= U+ PV (10-3)
and then into the Gibbs energy G
G= H– TS= U+ PV– TS (10-4)
The total exact differential of G yields
dG= –SdT+VdP (10-5)
which has the desirable property that the
independent variables Tand Pare easily
modified and controlled in an experiment.
This makes Gthe critical quantity
, which
has to be considered when comparing a
system at different states. While Pelton
(2001) focuses on the variation of Gas a
function of the chemical composition and
temperature, we will here investigate the
change in Gupon increasing pressure.
If a system is subject to changing pres-
sure at constant temperature, Eq. (10-5) re-
duces to
dG= VdP (10-6)
The change in the Gibbs energy on chang-
ing pressure is thus
(∂G/∂P)
T= +V (10-7)
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98. 10.2 Pressure-Driven Phase Transitions 661
Equation (10-7) states an intuitive trivial-
ity, i.e., that in order to minimize the Gibbs
energy upon increasing pressure, a given
system will reduce its volume. Volume re-
duction is thus the ultimate driving force of
structural change and phase transformation
as a reaction on increasing pressure. How-
ever, the way a volume is reduced can be
different, depending on the initial structure
and configurational entropy of the system
and also on the amount of pressure applied.
In the following we will focus on the three
most important mechanisms responsible
for structural changes at high pressure.
These are framework flexion, increase in
coordination number and pressure-induced
ordering.
10.2.1 Framework flexion
A number of technologically important
oxide materials (e.g., zeolites, perovskites)
can be viewed as being built of relatively
rigid corner-linked polyhedra forming a
rather flexible framework. These materials
are known to react on changing tempera-
ture and/or pressure mainly by flexion at
the polyhedral joints rather than by polyhe-
dral compression, i.e., bond length reduc-
tion (e.g., Hazen and Finger, 1978; Velde
and Besson, 1982; Hemley et al., 1994).
Silica (SiO
2) can be viewed as a typical ex-
ample of the behavior of framework struc-
tures under pressure. It has a rich phase di-
agram in P–Tspace. The ambient condi-
tion phase is
a-quartz, which consists of
interconnected spirals of corner-linked
SiO
4tetrahedra. Silica has two high-pres-
sure polymorphs, of which only coesite can
be viewed as a tetrahedral framework
structure. Its framework is characterized by
four-fold rings of tetrahedra connected to
chains, which in turn are packed via shared
tetrahedral corners to the densest known
tetrahedral framework. At ambient condi-
tions (coesite can be quenched to room
pressure), both structures exhibit nearly
ideal Si–O–Si angles of between 143° and
144°. The phase transition is thus structu-
rally characterized by building the tetrahe-
dral framework in coesite more densely
than in a-quartz. This can be demonstrated
by looking at the oxygen surroundings in
both structures: In a-quartz, each oxygen
has three oxygen neighbors at ~ 2.6 Å,
which represent the tetrahedral oxygen
neighbors. The closest oxygen atoms from
any other tetrahedron are found at a dis-
tance of ~ 3.5 Å. In coesite on the other
hand, there are also three oxygen neighbors
at 2.6 Å around each O, indicating the stiff
behavior of the SiO
4tetrahedrals. The next
nearest oxygen neighbors in coesite, how-
ever, are observed at a distance of only
3.0 – 3.2 Å (room pressure). Both structure
types react to applied pressure with a very
large decrease in the Si–O–Si angles (Fig.
10-1), while the Si–O bond lengths remain
more or less constant. Densification within
the stability field of quartz is thus achieved
by flexion of the framework through rigid
rotations of the SiO
4tetrahedra, resulting
in a decrease of the Si–O–Si angle from
144° to 125°. If this reaches a limit, denser
packing of the polyhedra is achieved by
framework reconstruction. Further densifi-
cation, however, eventually involves an in-
crease in the coordination number (see Sec.
10.2.2). Thus, a pure compression at RTor
a rapid (shock) compression of a-quartz
leads to an amorphization involving both
framework collapse and an increase in co-
ordination number (Hemley et al., 1994).
Pressure-induced amorphization seems to
be a common feature of many framework
structures, indicating a volume reduction
by a collapse of the framework. This de-
stroys the long-range order while still
largely maintaining the short-range order
of the primary building blocks. A similarwww.iran-mavad.com
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99. effect is also thought to be responsible for
the high-density form of amorphization of
ice (Mishima et al., 1984; see also Sec.
10.5.3.6).
The most open frameworks known in in-
organic chemistry are adopted by the zeolite
structure family. These structures exhibit
some peculiar effects upon compression.
Zeolite frameworks are characterized by
large cages and channels, which can be oc-
cupied by extra-framework cations or mole-
cules. As in silica, volume reduction on zeo-
lite can be most easily achieved by a reduc-
tion of the framework channels, which is
achieved through rigid body rotations of the
framework tetrahedra and thus bending of
the Si–O–Si angles. A study of the natural
zeolite natrolite for example showed that
compression leads to a continuous reduction
of the unit-cell volume without any structu-
ral phase transition up to ~ 7 GPa (Belitsky
et al., 1992). Above 7 GPa, the investigated
samples underwent amorphization, similar
to the transition observed for silica. This
amorphization again indicates the collapse
of the tetrahedral framework into a glassy
tetrahedral arrangement.
If the channels and holes in a framework
are large enough, we can even obtain the
seemingly paradox result of a negative
compressibility. This happens if the molec-
ular size of the pressure medium is small
enough to be squeezed into the channels.
Because the pressure medium seems to be
packed more densely within the zeolite
structure than in the fluid, the total volume
of the system (crystal plus pressure me-
dium) decreases. The volume of the crystal
alone, however, increases with applied
pressure. In a similar way, Hazen (1983)
observed compressional anomalies for zeo-
lite 4a, which exhibits different phase tran-
sitions depending on the pressure medium
used. All of the observed high-pressure
phases showed higher compressibilities at
high pressure.
A more frequent but still unusual phe-
nomenon observed upon compressing
framework structures is a negative linear or
areal compressibility. Materials with these
662 10 High Pressure Phase Transformations
Figure 10-1.Si–O–Si angles (solid symbols) and average Si–O bond lengths (open symbols) vs. pressure for
quartz (squares) and coesite (diamonds). Note the (within experimental errors) constant ·Si–OÒdistances with in-
creasing pressure, while the Si–O–Si angles show a significant negative correlation. This demonstrates that frame-
work structures (such as tetrahedral SiO
2polymorphs) adapt their volume through framework flexion. Data from
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100. 10.2 Pressure-Driven Phase Transitions 663
properties will expand in one or two di-
mensions upon hydrostatic compression.
As shown by Baughman et al. (1998), this
also implies a negative Poisson’s ratio, i.e.,
lateral contraction upon uniaxial compres-
sion. These authors also demonstrate that
properties such as a negative Poisson’s ra-
tio are in most cases linked to geometric
constraints in hinged framework structures.
Because such behavior can result in in-
creasing surface area upon increasing hy-
drostatic pressure, it is of potential interest
to material scientists. In principle it is con-
ceivable to manufacture these compounds
into porous composite material with zero
or negative volume compressibilities.
Another example of a structure family
whose phase transitions are characterized
by framework flexion are the perovskites.
This material, with the general formula
ABX
3, can be viewed as a stuffed deriva-
tive of WO
3-type structures. WO
3(BX
3) is
built up of a three-dimensional network of
corner-linked WO
6octahedra, the W (B)
cations occupying the corners of the cube-
shaped unit cell. In perovskite the center of
this cube is occupied by the A cations. In
the ideal structure, the B–X–B angle at the
connecting octahedral corners is 180°. As
shown by Glazer (1972), this octahedral
framework is susceptible to a number of
distortions which are characterized by rigid
octahedral rotations, thus pure framework
flexions. These distortions can even be ob-
served at ambient conditions depending on
the nature of the A and B cations. At ambi-
ent conditions, the geometry is controlled
by the relative size and thus the bonding re-
quirements of the cations involved. In a
similar way the effect of compression de-
pends on the relative size at ambient condi-
tions and the relative compressibilities of
the A and B cations. In most cases the A
cation is the more compressible unit, thus
leading to increasing polyhedral tilting
(i.e., framework flexion) with increasing
pressure. An example of this is the increas-
ing orthorhombic distortion of MgSiO
3
with increasing pressure (Fiquet et al.,
2000).
10.2.2 Increase in coordination number
Many crystal structures are best de-
scribed by a closest packing of anions with
some of the interstitial sites (two tetrahedra
and one octahedron per anion) occupied by
a cation. In this simple, but in many cases
very successful picture, the large and soft
anions are in contact with each other and
the small, more rigid cations are isolated
from each other and in contact only with
their surrounding anions. The number of
anions surrounding any given cation is de-
termined by the ratio of the “sizes” of the
cations and anions (for the problem of de-
fining the “size” of an atom see e.g., Ross
and Price (1997)). In such a close-packed
array of anions, rigid polyhedral rotation is
usually not able to accommodate a volume
reduction imposed by increasing pressure.
The only way to optimize the volume upon
increasing pressure is through more effi-
cient packing of the anions involved, thus
reducing the anion–anion distances within
the structure. This reduces the ‘size’ of the
anions without affecting the respective size
of the cations, which leads to a higher cat-
ion/anion size ratio. As a consequence, the
coordination number of the cations in-
creases. Such a coordination increase is
generally accompanied by a lengthening of
the cation–anion distances, which at first
glance may appear a surprising effect for a
high-pressure phase transition. However,
the lengthening of the first coordination
sphere is compensated by a shortening of
the second shell. It is therefore worthwhile
looking not only at the nearest neighbors of
a cation. When looking at both the first andwww.iran-mavad.com
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101. second coordination shells of a cation, a
coordination increase can be visualized as
making the bonding environment of a cat-
ion less distorted. Keeping this in mind, the
increase in coordination number can be
quantitatively rationalized by the distortion
theorem (Brown, 1992). The compression
of a close-packed array of anions leads first
to a reduction in the anion–anion distance
without rearranging the actual packing.
This reduces all cation–anion distances,
which leads to a strengthening of the indi-
vidual bonds. In the framework of Brown’s
bond valence approach, a general strength-
ening of cation–anion bonds around a
given atom induces an ‘overbonding’ of
the atoms, which in turn destabilizes the
structure. Because of the exponential rela-
tionship between bond strength and bond
length (Fig. 10-2), a set of equal bond
lengths will have a lower bond valence sum
than the same number of bonds with the
same average value, but different individ-
ual bond lengths (distorted arrangement).
An increase in coordination number thus
reduces the overbonding of the atoms in
the crystal by making the bonding environ-
ment around the atoms less distorted. A
very instructive example of this effect is
the aforementioned phase transition in sil-
ica (SiO
2) from coesite (4-coordinated Si,
2-coordinated O) to stishovite (6-coordi-
nated Si, 3-coordinated O). While the co-
esite structure is described by a very dense
packing of corner-linked SiO
4tetrahedra,
stishovite adopts the rutile structure, char-
acterized by chains of edge-sharing SiO
6
octahedra. Other prominent examples of
increasing coordination numbers at in-
creasing pressure are the well-known phase
transitions from the NaCl (6-fold) type
structure to the CsCl (8-fold) structure in
alkali chlorides or the high-pressure transi-
tions from zincblende (4-fold) to NaCl or
b-Sn (6-fold) in many AX semiconductors
(see Sec. 10.5.1).
10.2.3 Pressure-induced ordering
The potential effect of high pressure on
order–disorder phase transitions has only
recently been fully realized. The interested
reader is referred to the excellent and
thorough review by Hazen and Navrotsky
(1996). In this section we only give a brief
summary of the main features of pressure-
induced order–disorder phenomena.
Atoms on a given crystallographic site
can order with respect to their chemical
species, exact position, magnetic moment
664 10 High Pressure Phase Transformations
Figure 10-2.The exponential rela-
tionship between bond valence and
bond length leads to more regular
geometries at high pressure; High
pressure tends to destabilize struc-
tures through overbonding. For a
given average bond length, the aver-
age bond valence is higher (= more
overbonding) the more the individ-
ual bond lengths are different from
each other (= irregular geometry).www.iran-mavad.com
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102. 10.2 Pressure-Driven Phase Transitions 665
or electronic state. In the following, disor-
der with respect to chemical species is re-
ferred to as ‘substitutional disorder’,
whereas disorder on two neighboring sites
is called ‘structural disorder’. Changes in
any of these order parameters across a
phase transition are known to have an ef-
fect on the molar volume of the material
(e.g., Owen and Liu, 1947). Depending on
whether the molar volume is smaller for an
ordered or disordered state, pressure can
thus – in principle – promote ordering or
disordering effects. Nevertheless, such ef-
fects are very often hampered by reduced
diffusion rates at high pressure. It is there-
fore fair to say that at moderate tempera-
tures (1500 K), high pressure does not
necessarily induce order–disorder transi-
tions, but certainly supports such reactions.
The fast kinetics above ~1500 K, however,
allow rapid equilibration with respect to or-
dering and even in many cases inhibit
quenching of an ordering pattern stable at
high pressure and high temperature (Hazen
and Navrotsky, 1996).
Based on present data, the volume
change DV
dis= V
disordered– V
orderedof struc-
tural or substitutional disorder tends to be
positive, suggesting that high pressure fa-
vors an ordered arrangement (Hazen and
Navrotsky, 1996). This can be understood
by the fact that an ordered arrangement of
two atomic species of different ‘sizes’
leads to an alternation of ‘small’ and
‘large’ layers or rods (Fig. 10-3a). A disor-
dered arrangement, on the other hand,
forces each of the disordered sites to have
the apparent size of the largest atom shar-
ing this site (Fig. 10-3b). An ordering de-
pendence of the molar volume (and there-
fore a pressure dependence of the ordering)
can also be observed in flexible framework
structures such as feldspars. At first glance,
the molar volume of such structures should
not be critically correlated with the size of
the cations, because in these structures
most of the volume change induced by or-
dering or pressure can be accommodated
by the intra-polyhedral angles (Sec.
10.2.1). The observed volume changes,
however, can be understood on the basis of
the variation of intra-polyhedral angles, de-
pending on the cation species occupying
the respective polyhedra (e.g., Geisinger et
al., 1985). Simple geometric considera-
tions show that varying distributions of a
given set of angles in space leads to differ-
ent enclosed volumes (Fig. 10-4).
Figure 10-3.Schematic drawing
to illustrate a possible mecha-
nism of pressure-induced order-
ing. Two sets of balls of different
sizes occupy (a) a smaller vol-
ume in an ordered arrangement
than (b) the same number of
balls in a disordered distribution.www.iran-mavad.com
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103. The largest volume changes are ob-
served for substitutional ordering pro-
cesses. A
4+
B
2
2+O
4spinels, for example,
show a difference in molar volume of up to
3.5% between the normal (fully ordered)
and the inverse (disordered on the octahe-
dral site) modifications (Hazen and Yang,
1999). The pressure behavior of the spinel
family is especially interesting, because
they show both pressure-induced disorder-
ing as well as pressure-induced ordering
for one and the same structure type de-
pending on the chemical species involved
(Wittlinger et al., 1998; Hazen and Yang,
1999).
The rather scarce data on pressure de-
pendence of charge distribution seems to
indicate that charge ordering tends to be
suppressed by high pressure. NaV
2O
5for
example shows a charge-ordering transi-
tion around 35 K at ambient pressure. This
transition seems to be shifted to lower tem-
peratures at higher pressures and disap-
pears completely around 1 GPa (Ohwada et
al., 1999). This suggests that in the case of
NaV
2O
5, high pressure induces a charge-
disorder phase transition. In a similar way,
the application of pressure to Sm
4Bi
3shifts
the charge-ordering transition to lower
pressures and eventually even induces an
iso-structural phase transition where the
mixed valence compound Sm
3
2+Sm
3+
Bi
3
3–
changes to a purely 3-valent material
Sm
4
3+Bi
3
4–(Ochiai et al., 1985).
Another technically interesting phenom-
enon that is connected to the pressure de-
pendence of ordering and correlation of
charge carriers is the well-documented
pressure dependence of T
cin certain ce-
ramic superconductors (e.g., Acha et al.,
1997; Han et al., 1997). Although neither
the superconductivity nor its relationship
to high pressure is fully understood in ox-
ide materials, it is justified to assume that
the observed strong shift in T
cwith increas-
ing pressure is connected to a subtle inter-
play between pressure-induced structural
distortions, orbital overlaps and charge-
carrier distribution.
10.3 Generating High Pressure
The technology of pressure cells com-
patible with in situ experiments for mate-
rial characterization has experienced tre-
666 10 High Pressure Phase Transformations
Figure 10-4.Schematic two-
dimensional sketch to demon-
strate how the distribution of a
given set of rigid triangles
(polyhedra in 3-d) affects the
area (volume in 3-d) enclosed.
The area between the triangles
in (a) is 2.450, while the trian-
gles in (b) enclose an area of
2.414 (after Hazen and Navrot-
sky, 1996).www.iran-mavad.com
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104. 10.3 Generating High Pressure 667
mendous development since the ground-
breaking work of Bridgman. Consequently,
there is a very comprehensive and vast lit-
erature on this subject to which the more
committed reader is referred (Miletich et
al., 2000; Holzapfel, 1997; Eremets, 1996;
Ahrens, 1987). Only a brief overview will
be given in this section.
There is a huge variety of different tech-
niques for generating pressures. They can
initially be divided into static methods and
dynamic techniques.
10.3.1 Dynamic pressure generation
The highest pressures (10
2
–10
3
GPa)
can be obtained using dynamic shock-wave
generation. This is achieved by means of
explosives or by a projectile that is acceler-
ated toward the target with a gas gun (Ah-
rens, 1980, 1987). The shock-wave tech-
nique was originally developed in the mid-
1950s at Los Alamos, USA, in the course
of the development of atomic bombs
(Walsh and Christian, 1955). In its simplest
case, the impact of a projectile on the target
produces a uniaxial shock wave. The shock
wave passes through the sample at shock
velocity U
s. The sample itself is acceler-
ated to the sample velocity U
p, and U
sand
U
p, together with the temperature, are the
quantities measured during a shock-wave
experiment. The velocities are usually de-
termined by measuring entrance and exit
times of the shock wave. For samples of
a few millimeters in length, the time to
be measured is in the range of 10
–1
to
~<10
1
µs. Ignoring the yield strength of
the solid (which is justified at the high
shock pressures encountered during a
shock-wave experiment), the material be-
haves as a fluid. In such an experiment, the
volume decreases from V
0to V
H, the tem-
perature increases from T
0to T
Hand the
pressure from P
0to P
H, while the internal
energy rises from E
0to E
H. The Rankine–
Hugoniot relations combine these quanti-
ties to (
r
0= density at ambient conditions):
V
H= V
0(U
S– U
P)/U
S (10-8)
P
H= r
0U
SU
P (10-9)
E
H– E
0= (V
0– V
H)P
H/2 (10-10)
The measurements of U
Sand U
Pgive the
quantities on the left of Eqs. (10-8) to (10-
10) for one experiment. Various experi-
ments at different strengths of explosion or
different velocities of the projectile pro-
duce different points, forming the Hugon-
iot curve describing the Hugoniot equa-
tions of state V
H(P
H). In order to reduce the
Hugoniot equations of state into an isother-
mal equation of state, careful thermody-
namic corrections have to be applied
(Poirier, 1991). A material passed by a
shock wave usually displays various
stages. Up to a pressure of the Hugoniot
elastic limit (HEL) (0.2 to 20 GPa), the
sample behaves elastically, corresponding
to the propagation of the longitudinal
shock wave. Above the HEL, plastic defor-
mation of the material occurs, giving rise
to the fluid-like behavior that creates the
Hugoniot curves. If a material undergoes a
phase transition, this is readily observed as
changes of slope in the Hugoniot curves,
separated by a mixed-phase regime (Fig.
10-5).
In this sense, in a shock-wave experi-
ment the material can only be investigated
by the difference in U
Pand U
S, which both
depend on the volume reduction of the
sample. The accuracy of these measure-
ments becomes critical once the volume re-
ductions are small at very high pressures.
Third-generation synchrotron radiation
sources or even free-electron lasers of the
future may offer improved possibilities.
The increase in X-ray flux of these X-ray
sources may allow for stroboscopic X-raywww.iran-mavad.com
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105. diffraction in the microsecond regime. This
would enable structural information to be
extracted of a material under extreme
dynamic pressure. Today, shock-wave ex-
periments are highly complementary to
static high-pressure experiments. Undoubt-
edly, one of their most crucial roles is the
creation of an equation of state up to ex-
treme pressures without any additional
pressure standard. This provides invaluable
anchor points for all pressure scales cur-
rently used in static high-pressure experi-
ments (Mao et al., 1978; Jamieson et al.,
1980).
10.3.2 Static pressure devices
Devices to generate static pressures that
are sustainable for an a prioriindefinite
time can be divided into ‘large-volume
presses’ and ‘diamond anvil cells’. The pri-
mary difference between these two fami-
lies of devices lies in the volume of mate-
rial subjected to high pressure and conse-
quently the maximum pressure attainable.
For large-volume devices the compressed
volume lies between 1 mm
3
and 1 cm
3
.
This allows maximum pressures of around
20 GPa to be obtained. Diamond anvil
cells, in contrast, enclose a volume of
<10
–3
mm
3
, i.e., only a few pico-liters.
These devices are capable of maximal
pressures up to 200 to 500 GPa.
103.2.1 Large-volume presses
Historically, the first large-volume cells
were hydraulic presses as built by Bridg-
man. He was able to compress fluids up to
10 GPa. In modern high-pressure research,
hydraulic devices are limited to maximal
pressures between 1 and 2 GPa and will
therefore not be discussed any further in
this chapter. Presses operating with solid
pressure media up to 50 GPa are the most
important tools in modern materials sci-
ence for the synthesis of materials at simul-
taneously high pressure and high tempera-
ture. Increasingly, large-volume devices are
also used for in situstudies in combination
with synchrotron and neutron radiation.
Piston-Cylinder systems are the simplest
design for compressing material. They rely
on a cylinder acting as the sample chamber.
The sample is compressed by a piston fit-
ted within the cylinder. Closed-end cylin-
668 10 High Pressure Phase Transformations
Figure 10-5.Hugoniot curve of a
hypothetical material undergoing a
phase transformation at high pres-
sure. The low-pressure phase ap-
proaches its hydrostatic behavior af-
ter passing the Hugoniot elastic limit
(HEL). The existence of a shock-in-
duced high-pressure phase is seen
by the onset of the mixed-phase re-
gime. The hydrostat of the high-
pressure phase can be recognized in
the high-pressure phase regime (af-
ter Ahrens, 1980).www.iran-mavad.com
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106. 10.3 Generating High Pressure 669
ders are closed on one side and the sample
is compressed with one piston from the
other side. Open-end cylinders rely on two
opposite pistons. The maximal pressure of
5 GPa attained by piston-cylinder devices
is generally limited by the yield strength of
the cylinder. Various techniques exist to in-
crease this yield strength. In a technique
called frettaging the cylinder is prepared
such that in its default state, the outer part
retains a residual tensile stress and the in-
ner part a compressive stress. Adding the
work stress leads to a partial cancellation
of these stresses, thus allowing for the ap-
plication of higher loads to the sample.
Pre-loading the cylinder is another way to
increase the yield strength of the cylinder
to higher values. Pre-load can be achieved
by using a soft but incompressible material
(lead) as the inner cylinder material and
by initially overstraining it into its flow re-
gime. Another method relies on the appli-
cation of an external load on the cylinder
simultaneously with the application of the
load on the piston. Alternatively, cylinders
can be reinforced by winding a strong wire
under tension around the cylinder. The
most straightforward way to reinforce a
cylinder is by increasing the ratio of its di-
ameter to length. If this is pursued conse-
quently, we arrive at two conical pistons
compressing a sample contained in a girdle
(Fig. 10-6a). A modification of the girdle
design is obtained by optimizing the coni-
cal shape of the anvils. A cycloid shape of
the pistons, as shown in Fig. 10-6b, en-
sures the optimal compromise ensuring
sufficient cylinder support at low pressure
while maintaining a reasonable stroke in
the high-pressure regime. This ‘belt
design’ is by far the most widely used high-
pressure apparatus for materials synthesis
and is frequently applied in industry for the
synthesis of super-hard material. The max-
imum pressure achieved by belt devices is
around 10 GPa. Such devices have also
been optimized with respect to sample vol-
ume; a flat belt apparatus constructed by
Fukunaga et al. (1987) was capable of
compressing a sample of 125 ml up to pres-
sures of 5 – 7 GPa.
The disadvantage of piston-cylinder as-
semblies is the intrinsic opacity of both pis-
ton and cylinder, thus inhibiting most in
Figure 10-6.Schematic drawing of (a) a girdle-type press and (b) a belt apparatus. The shaded area corre-
sponds to gasket material. S shows the sample position. Note how consequent cutting of stress amplifying cor-
ners on the girdle and decrease of length-to-width ratio on the piston lead from a girdle press to a belt design.www.iran-mavad.com
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107. situobservations. This problem is allevi-
ated by opposed anvil systems which oper-
ate with a gasketing system instead of a
cylinder. The very first example of this
type was built by Bridgman (1952). A log-
ical extension of the Bridgman design are
profiled Bridgman anvils as first proposed
by Ivanov and coworkers in the 1960s
(e.g., Ivanov et al., 1991) leading to tor-
oidal anvils (Fig. 10-7). This principle was
optimized most consequently in the
Paris–Edinburgh cell design (Besson et al.,
1992). The advantage of X-ray and neutron
transparent gasketing has been exploited to
the maximum in this cell. This is the reason
why this cell design has advanced to the
state-of-the-art model for in situhigh-pres-
sure neutron diffraction and is also em-
ployed for X-ray experiments. A develop-
ment of the Los Alamos neutron group
pushes the limits of Pand Tattainable with
toroidal anvils up to 50 GPa and 3000 K,
thus enabling in situneutron studies up to
these conditions.
A quite different approach to compress-
ing large volumes up to pressures of
30 GPa was first developed by researchers
in Japan using multi-anvil designs (Aki-
moto, 1987). A very successful product is
the DIA-type cubic anvil press. In this de-
vice, six tungsten carbide (WC) or sintered
diamond anvils are arranged parallel to the
faces of a cube that encloses the cell as-
sembly. A ram exerts an axial force, which
is also translated into an equatorial com-
pression through wedges attached to the
four equatorial anvils. Although initially
designed for synthesis experiments, vari-
ous models of this type have been used
in recent years at synchrotron radiation
sources for in situX-ray diffraction studies
in Japan (Photon factory and Spring-8),
Brookhaven (NSLS) and Hamburg
(DESY). This is possible because the small
gaps between the individual anvils allow
for entry and exit of X-rays, and the cell as-
sembly containing the sample is made of X-
ray transparent material (BN, epoxy). With
tapered anvils made of sintered diamond,
maximal pressures of 20 – 25 GPa at tem-
peratures around 1000 °C can be attained.
A rather different geometry in multi-an-
vil technology is the split-sphere design
(Liebermann and Wang, 1992) pioneered at
Okayama University by Ito. It consists of a
uniaxial ram applying force on a split
sphere (first stage) that contains a cube-
shaped cavity with the body diagonal along
the axis of the ram. The cavity is occupied
by a cube built of eight WC cubes with
truncated corners. The truncated corners in
turn form an octahedral cavity that hosts
the cell assembly. The cell assembly is usu-
ally a MgO octahedron with a Pt capsule
containing the sample in its center. A vari-
ation of this design is the split-cylinder de-
sign in which the sphere of the first stage is
replaced by a cylinder. Multi-anvil presses
670 10 High Pressure Phase Transformations
Figure 10-7.The principle of
the toroidal anvil. The cut is
circular symmetric around the
vertical axis of the plot. The
sample (S) is surrounded by a
washer within a toroidal belt
containing gasket material. This
geometry prevents the sample
from being extruded, thus creat-
ing a hydrostatic pressure de-
spite the uniaxial force applied.www.iran-mavad.com
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108. 10.3 Generating High Pressure 671
based on the split-sphere approach are also
used as in situdevices at the ‘Synchrotron
Radiation Source’ in Daresbury (UK) and
the ‘Advanced Photon Source’ at Argonne
National Laboratory (USA).
However, the determination of pressure
is not straightforward. In principle, pres-
sure can be calculated by dividing the force
by the area on which it is acting. In practice
this is not applicable, mainly because of
the unpredictable friction losses and illde-
fined compressibilities of the cell assem-
bly. Therefore, pressure has to be deter-
mined through a calibration procedure.
When in situdiffraction techniques are im-
possible, this can be done by determining
phase transitions revealed through changes
in resistivity of metals and semiconductors
(e.g., Bi, Ba, ZnS, GaAs, GaP). The re-
spective high-pressure phase transitions
are determined in hydraulic pressure de-
vices whose pressure can be directly meas-
ured by a pressure gauge. If in situobserva-
tion is possible, the equations of state of
materials such as NaCl, Cu, Mo, Ag and
Pd, which have been determined up to very
high pressures through shock-wave experi-
ments, can be applied to calibrate the press.
Because the reproducibility for a given
press and cell assembly is very high, pres-
sure of subsequent experiments can be de-
termined from the force applied on the cell
assembly.
The strength of large-volume devices
lies in their potential for material synthesis
at simultaneously high temperature and
high pressure. The relatively large size of
the compressed volume (~ 0.1–1 cm
3
) al-
lows inclusion of a heater (usually cylindri-
cal graphite or LaCrO
3resistance heaters)
as well as thermocouples, which enable
pressure and temperature to be combined
in a very controlled way. Due to the large
volume, these devices are limited to maxi-
mal pressures of 50 GPa at the very best.
10.3.2.2 Diamond anvil cells
Experiments at extreme pressures, which
are of interest not only to geophysicists and
planetologists, but also to physicists and
chemists studying, for example, solidifica-
tion and metallization of ‘gases’, can be
achieved using diamond anvil cells (DAC).
Diamond anvil cells are in principle very
small opposed-anvil devices of Bridgman
type. The anvils are made of diamond sin-
gle crystals, shaped in the brilliant cut with
the bottom tip truncated to form the anvil
surface (culet). This simple design allows
the special properties of diamonds to be
used in two ways. First, the extreme hard-
ness of diamonds allows very high pressure
to be generated. Pressures attained depend
of course on the size of the culet. Maximal
pressures of 500 GPa have been reported
(Xu et al., 1986) and pressures between
100 and 200 GPa can be reliably repro-
duced. The second advantage of using
single crystalline diamonds as pressure
anvils is the high transparency of diamond
for almost the entire electromagnetic spec-
trum. This allows us not only to easily
observe samples under high pressure, but
also to probe them with spectroscopic
methods as well as X-ray diffraction
(Sec. 10.4). The strength of DACs is thus
their huge range of pressure combined with
the ease of performing in situexperiments,
and this all with a device of the size of a
fist, which is also easy to operate by non-
specialists.
The very first DAC was constructed us-
ing a big (8 ct) gem-quality diamond (taken
from smugglers by the US government and
donated to the US National Bureau of Stan-
dards) in which a hole was drilled (Jamie-
son, 1957; Lawson and Tang, 1950). Pres-
sure on the sample in the hole was applied
via a piston (piano string) pressed on the
sample. The limited pressure range obtain-www.iran-mavad.com
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109. able with this approach was painfully real-
ized when the 8 ct diamond was crushed
during an experiment. As a consequence,
the opposed-anvil geometry was developed
(Weir et al., 1959; Jamieson et al., 1959),
first by simply squeezing a powdered sam-
ple between the culets of two opposed an-
vils and later by introducing the gasketing
technique (Van Valkenburg, 1964). Its prin-
ciple has remained unchanged since its in-
vention and is as simple as it is efficient
(Fig. 10-8). A hole about 100 to 200mm in
diameter is drilled in a metal foil (Fe, W,
Re). This hole serves as sample chamber
and is filled with the sample (powder or
single crystal) and a pressure medium (al-
cohol, liquid gas). The pressure medium is
compressed through an axial force exerted
by the diamonds on the gasket. The gasket
seals the sample chamber and at the same
time transforms part of the axial pressure
into an equatorial pressure through the
plastic deformation of the gasket material.
In this way the pressure medium is com-
pressed isotropically and therefore trans-
mits a hydrostatic pressure on the sample.
This design was popularized for materials
research by Bassett (Merrill and Bassett,
1974) and Mao (Mao and Bell, 1975). Fur-
ther details on the technology of DACs and
also on its various modifications and devel-
opments are given in excellent reviews by
Hazen and Finger (1982) and Miletich et
al. (2000).
As for the large volume devices, the ac-
curate determination of pressure in a dia-
mond anvil cell is a difficult issue. The
most accurate values are obtained by add-
ing an internal standard to the sample
whose equation of state is known with suf-
ficient precision to relate its diffraction
pattern to a pressure value (e.g., Angel et
al., 1997). A very convenient and popular
alternative, albeit not quite as accurate, is
the exploitation of the pressure shift of flu-
orescence lines. The most frequently used
fluorescence is the R1 line of ruby (Mao et
al., 1978). This method again benefits from
the transparency of the diamond high-pres-
sure windows. A ruby, which is packed to-
gether with the sample into the gasket hole,
is illuminated with a green or blue laser
that induces a red fluorescence line. The
wavelength of this line depends on pres-
672 10 High Pressure Phase Transformations
Figure 10-8.Sketch illustrating the prin-
ciple of the diamond anvil cell. The axial
force applied to the diamonds is partly
translated into a plastic deformation of
the gasket, which results in a circular
symmetric equatorial force. This ensures
quasi-hydrostatic conditions at the sample
(S), which is embedded in a pressure me-
dium.www.iran-mavad.com
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110. 10.4 Probing Phase Transformations in Materials at High Pressure 673
sure (dl/dP= 0.37 nm GPa
–1
) and can thus
be used to determine the pressure within
the sample chamber. More recent develop-
ments of this approach use different fluo-
rescence lines with various pressure and
temperature dependence in order to simul-
taneously determine pressure and tempera-
ture in the sample chamber (e.g., Datchi et
al., 1997).
Because of the very small volumes of the
sample chamber (~ 0.001 mm
3
), combining
pressure and temperature in a diamond an-
vil cell is a difficult task. In principle there
are two different approaches, namely exter-
nal and internal heating. With external
heating, the whole sample chamber, includ-
ing diamonds, is enclosed in a resistance
heater (e.g., Hazen and Finger, 1982; Bas-
sett et al., 1993) that heats the entire assem-
bly consisting of gasket, diamond anvils
and sample. Temperature is measured via
thermocouples attached to the outer dia-
mond facets, assuming that the high ther-
mal conductivity of the diamonds allows
for only a very small temperature gradient
between sample and outer diamond facets.
This technique has consequently been opti-
mized by using the gasket material itself as
a resistance heater, thus minimizing the
heated volume and thermal gradients (Du-
brovinsky et al., 1997). An alternative ap-
proach is the use of an infrared laser beam,
which is focused through the (IR-laser
transparent) diamonds onto the sample,
where it is absorbed and thus transformed
into heat. This technique was again pio-
neered by Bassett (e.g., Bassett and Ming,
1972) and then further developed and opti-
mized by Boehler (Boehler and Chopelas,
1991) and Fiquet and Andrault (Fiquet et
al., 1994). Measuring temperature with this
technique is even more difficult because it
explicitly assumes that only the sample is
heated up. The only way to obtain a quanti-
tative estimate of the sample temperature
is by measuring the black-body radiation
of the glowing sample and fitting it to
Planck‘s spectral function (e.g., Bassett
and Weathers, 1987). When doing HP-HT
experiments with laser heating we should
bear in mind the possibility of thermal
pressure, i.e., the increase of pressure in
the heated area, while the pressure deter-
mined by a ruby chip outside the hot-spot
remains constant (Andrault et al., 1996).
10.4 Probing Phase
Transformations in Materials
at High Pressure
A very important aspect when doing ex-
periments on high-pressure phase transfor-
mations is the possibility of investigating
the phase transformation in situat condi-
tions of high pressure and possibly simulta-
neously high temperature. As mentioned
above, this is much easier to do in static ex-
periments than with dynamic shock-wave
techniques. This is mainly due to the very
short time that is available in a shock-wave
experiment. We will therefore focus on
static experiments in the following. Among
the static experiments, the transparency of
the diamond pressure windows in a DAC
allows for much more versatile experimen-
tal techniques compared with large-volume
presses. For large-volume experiments, di-
rect observation of the sample is limited to
either transport properties (e.g., electric re-
sistance) or, if using some sort of electro-
magnetic radiation as a probe, severe com-
promises in signal-to-background and ac-
cessible space have to be accepted. Never-
theless, the range of in situ techniques ap-
plicable to both DACs and large-volume
devices has grown considerably and con-
tinues to expand. There is an extensive spe-
cialized literature on each of the various
techniques. We will give here a brief over-www.iran-mavad.com
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111. view on and introduction to the available
techniques, with references to the more de-
tailed literature. Many techniques can be
used to measure different physical proper-
ties and vice versa . In the following we will
differentiate by the technique rather than
by the measured property.
10.4.1 Volumetric techniques
The first in situobservations in large-
volume cells were pioneered by Bridgman
(e.g., Bridgman, 1940) on piston-cylinder
devices. He made volumetric measure-
ments by simply measuring the stroke of
the piston as a function of force (pressure)
applied to the sample. This technique obvi-
ously assumes that deformation of the pis-
ton and cylinder, as well as leakage, can be
neglected. The accuracy in V/V
0obtained
through this method can be fine-tuned to
about 1 in 1000 (Anderson and Swenson,
1984). In general, the accuracy is limited to
lower values by deformation of the piston
and the cylinder. Compressibilities are
therefore most often and much more accu-
rately determined using diffraction tech-
niques (see Sec. 10.4.4). Prior to the avail-
ability of in situ diffraction techniques,
however, volumetric measurements were
the only way of measuring compressibil-
ities as a function of pressure, and they pro-
vided extremely valuable data.
10.4.2 Spectroscopic techniques
There are a variety of spectroscopic
techniques making use of various segments
of the electromagnetic spectrum. Although
the transparency of diamonds makes DACs
a very obvious tool for spectroscopic tech-
niques, they are by no means limited to
them, but – due to the advent of brilliant
synchrotron radiation sources – are in-
creasingly also applied in combination
with large-volume presses.
10.4.2.1 Microscopy
A rather cheap, but in many cases highly
efficient ‘spectroscopic’ device is the hu-
man eye. Visual observation of a sample
under high pressure can be a very useful
and sensitive tool for observing a phase
transformation and pinpointing it in P–T
space. An example of this has been de-
scribed by Arlt et al. (2000), where the
pressure dependence of a high-temperature
phase transition in Mn-pyroxenes turned
out to be difficult to determine using dif-
fraction techniques, but could be optically
observed through the discontinuous change
in birefringence.
10.4.2.2 Raman and infrared
spectroscopy
The most popular spectroscopic tech-
niques applied in high-pressure studies are
Raman and infrared (IR) spectroscopies.
They both probe the vibrational properties
of the material under investigation. Be-
cause lattice vibrations strongly depend on
the topology of the chemical bonds of a
substance, these spectroscopic techniques
are very sensitive to phase transformations.
Vibrational modes involving a dipole
change can be excited by absorbing an in-
frared photon giving rise to an absorption
band in the infrared. The energies of these
lattice modes are thus of the same magni-
tude as the energy of the IR photons. If the
photon energy is much higher, the interac-
tion of the photon with the lattice can in-
duce a vibrational mode, where the energy
of the lattice mode is transferred from the
photon to the lattice. This decreases the en-
ergy of the photon and therefore induces a
wavelength shift on the scattered photons
(Raman spectroscopy). Because energy can
be transferred in both ways (to and from
the lattice), the respective wavelength shift
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112. 10.4 Probing Phase Transformations in Materials at High Pressure 675
can be positive or negative. By measuring
the wavelength shifts of transmitted laser
light of a wavelength around 500 nm, a
characteristic fingerprint of the vibrational
properties of a given substance is obtained.
Again, this fingerprint critically depends
on bond strengths and the structural config-
uration and is therefore ideal for detecting
phase transformations. While in general
Raman shifts are measured for the optical
branches of the vibrational spectrum, very
carefully designed experiments are able to
determine the minute frequency shifts
caused by the acoustic branches, too. Such
Brillouin scattering experiments are very
interesting, because they allow the full
elastic tensor to be measured as a function
of pressure (e.g., Sinogeikin and Bass,
1999). The elastic tensor is a material prop-
erty that is useful in its own right. Its strong
dependence on structural parameters
makes it a very sensitive probe for detect-
ing and investigating phase transforma-
tions (Carpenter and Salje, 1998). A very
instructive and comprehensive overview of
high-pressure specific problems and appli-
cations of IR and Raman spectroscopy is
given by Gillet et al. (1998).
10.4.2.3 Mössbauer spectroscopy
Mössbauer spectroscopy has become in-
creasingly popular in materials science to
probe site-dependent distributions of
charge and magnetic moment. Its applica-
tion to high pressure is limited to DAC ex-
periments (Pasternak and Taylor, 1996;
McCammon, 2000). In a classical Möss-
bauer experiment, the radiation emitted
by the
g-source lifts the sample nuclei into
an excited state through an absorption
event. During re-emission, a fraction of the
g-quanta is emitted without recoil on the
lattice (recoil-free fraction) and can thus be
reabsorbed by a nucleus in the same struc-
tural environment (resonant absorption)
while lifting it into an excited state. The
energy of the excited state of a given nu-
cleus is a function of its structural environ-
ment, and therefore varies between indi-
vidual substances. This energy shift (iso-
mershift) is another characteristic finger-
print for a given structural state as well as
for the electronic charge of the nucleus.
The isomer shift can be measured by alter-
ing the relative energy of the
g-quanta
through the Doppler effect caused by rela-
tive movements of sample and source. The
g-quanta of the most popular Mössbauer
nuclei (
57
Fe
26) are in the range of 14.4 keV
and are thus strongly absorbed even by di-
amonds. It is for this reason that the first
high-pressure Mössbauer experiments
were performed using more exotic
Mössbauer nuclei such as
153
Eu,
129
I and
170
Yb, which have higher g-energies.
These experiments helped answer some
interesting questions about the physics of
magnetic materials (e.g., Pasternak et al.,
1986; Abd-Elmeguid et al., 1980). The de-
velopment of especially miniaturized dia-
mond anvil cells (e.g., Pasternak and Tay-
lor, 1990, 1996) together with advances in
detector technology also allowed success-
ful Mössbauer experiments to be per-
formed on
57
Fe nuclei (e.g., McCammon et
al., 1998). While traditional Mössbauer ex-
periments make use of the energy structure
of the emitted g-rays, modern synchrotron
sources with their time-pulsed radiation
also allow us to exploit the time structure of
a Mössbauer event. As mentioned above,
a Mössbauer event involves the absorption
of a
g-quantum by lifting the Mössbauer
nucleus in an excited state. This state lasts
for a time span in the range of 100 ns be-
fore it decays and emits the scattered radia-
tion. Because individual synchrotron X-ray
bursts can be gated into a time interval in
the range of 100 ps with 1ms
–1
repetitionwww.iran-mavad.com
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113. rates, the actual decay event can be sepa-
rated from the primary synchrotron radia-
tion. This allows the individual decay
events and their interaction with the nu-
clear and magnetic lattice to be investi-
gated directly. In such a ‘nuclear resonant
scattering’ experiment, the emitted
g-rays
of the individual nuclei in the material have
different wavelengths owing to the hyper-
fine interaction of the magnetic
57
Fe nu-
cleus. The different wavelengths of radia-
tion emitted from different nuclei cause a
quantum-beat oscillation pattern, which in
turn yields information on the material in-
vestigated (e.g., Smirnov, 1999). The high
brilliance of synchrotron radiation sources,
which are required for such experiments,
also makes nuclear forward scattering
much easier to apply in combination with
diamond anvil cells. High-pressure experi-
ments using nuclear forward scattering as a
probe to investigate structural and mag-
netic phase transitions have therefore
quickly become very popular (e.g., Nasu,
1996; Lubbers et al., 1999).
10.4.2.4 X-Ray absorption spectroscopy
The high X-ray brilliance of modern syn-
chrotron radiation facilities not only revo-
lutionized high-pressure diffraction (see
Sec. 10.4.4), but also provided the opportu-
nity to apply X-ray absorption spectros-
copy (XAS) methods in high-pressure
research. In particular, developments on
large-volume presses (Paris–Edinburgh)
which were originally built for neutron dif-
fraction, proved to be very useful for in situ
XAS experiments. Most popular among the
high-pressure XAS techniques are the ‘ex-
tended X-ray absorption fine structure’
(EXAFS) experiments (i.e., Katayama et
al., 1997). The principle of EXAFS is well
known and has been described since the
early days of quantum mechanics (see
Brown et al. (1988) for a review). If an in-
cident X-ray photon hitting an atom in the
sample has an energy equal to the energy
difference between the ground state and the
excited state of a core electron, it will be
absorbed by the atom while lifting the core
electron to an excited state. Photons at this
energy have a high probability of being ab-
sorbed by the sample, which leads to char-
acteristic absorption edges in the transmit-
ted X-ray spectrum. Core electrons in their
excited state (photoelectrons) are delocal-
ized from their parent atom and can thus
interact with the intermediate surroundings
of the atom as well as with other photoelec-
trons, leading to an oscillatory contribu-
tion in the X-ray spectrum in the vicinity of
an absorption edge. It is this oscillatory
part that is extracted from an EXAFS ex-
periment. It is dependent on the immediate
surroundings of an atom and therefore con-
tains information about the local neighbor-
hood of an atom in a solid or a liquid.
When photoelectrons are recaptured by an
atom, they will emit characteristic fluores-
cence radiation which by itself can again
be used as a probe for characterizing a ma-
terial. The advantage of XAS methods in
comparison with diffraction methods is
their sensitivity to local and short-range ef-
fects. This makes them very powerful for
the investigation of phase transformations,
not only in crystalline solids but also in
amorphous solids and liquids (e.g., Buon-
tempo et al., 1998).
10.4.2.5 NMR
A final spectroscopic tool that can be ap-
plied to samples under high pressure is
NMR. An NMR experiment exploits the
interaction between the magnetic moment
and the spin of the nucleus on the one hand
and a static magnetic field disturbed by
pulses of a radio frequency field on the
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114. 10.4 Probing Phase Transformations in Materials at High Pressure 677
other. Introductory texts on NMR are avail-
able, e.g., by Akitt (1983) or Kirkpatrick
(1988). The spin of a nucleus leads to a
precession motion of its magnetic moment
if placed in a static magnetic field. A mag-
netic perturbation in the radio frequency
range of the static field causes a tilting of
the precession axis. The time the nucleus
needs to revert its precession axis into its
static position depends on the nature of the
nucleus itself as well as its immediate sur-
roundings in the crystalline or amorphous
host. NMR, similar to XAS techniques, is
therefore a useful probe for investigating
short-range phenomena. Alternatively, the
analysis of NMR spin echoes also allows
information on viscosity and self-diffusion
rates in liquids to be extracted.
The application of NMR to high pressure
is an experimental challenge because it has
to deal with both the inherently small vol-
umes encountered in high-pressure experi-
ments and the magnetic susceptibility of
most materials suitable for construction of
high-pressure devices. Nevertheless, the
first high-pressure NMR experiment was
performed as early as 1954, in combination
with a Bridgman-type press (Benedek and
Purcell, 1954). The press was almost en-
tirely made from a non-magnetic Be–Cu al-
loy. Combining NMR and DAC is even
more difficult because of the extremely
small sample volumes. However, Bertani et
al. (1992) developed an NMR–DAC, again
made of Be–Cu alloy. This DAC is compat-
ible with a cylindrical cryostat and is thus
able to combine high pressure and low tem-
perature, allowing for the study of the pres-
sure and temperature dependence of the
Knight shift in solids (e.g., Kluthe et al.,
1996). An NMR set-up for the investiga-
tion of gases and liquids at pressures up to
20 bar has been developed by Woelk and
Bargon (1992).
10.4.3 Ultrasonic sound velocity
Measuring sound velocities at high pres-
sure is of great interest, mainly for seismol-
ogists who use this to correlate seismic
models with mineralogical models of the
earth. The importance of this method for
materials science is not quite so obvious al-
though in principle, discontinuities in the
change in sound velocity at high pressure
can be used to detect phase transformations
and to characterize them through the elastic
tensor. A simple pulse-transmission or
pulse-echo method can be combined with a
piston-cylinder apparatus in order to mea-
sure velocities at pressures up to about
1 GPa. A more sophisticated approach is
ultrasonic interferometry, which can be
combined with a split-sphere apparatus. In
this technique, two phase-coherent pulses,
separated in time by the approximate return
travel time in the sample, are applied to a
cell assembly under pressure, which is at-
tached to a buffer rod. When combining
this technique with a large-volume press,
the anvils are simultaneously acting as buf-
fer rods. For two consecutive pulses, the
echo of the first pulse from the far end of
the sample overlaps with the echo of the
second pulse from the sample–buffer inter-
face. The interference caused by this over-
lap leads to a modification of the amplitude
of the resultant signal. This amplitude in
turn can be modified by changing the car-
rier frequency, causing a beat pattern as a
function of frequency. The difference on
the frequency scale between two succes-
sive interference extremes yields the sound
velocity, if the dimensions of the sample
are known. This method has an advantage
over more traditional pulse-transmission
techniques, in that it involves only one
interface, reducing the possible coupling
problems for the acoustic signals. Owing to
the applied high pressure, mechanical con-www.iran-mavad.com
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115. tact and thus coupling between the materi-
als forming the interface is further en-
hanced. A more thorough introduction to
this technique is given by Rigden et al.
(1992) and Li et al. (1998). There have
been recent efforts to combine ultrasonic
interferometry with diamond anvil cells. In
this case, however, because of the very
small sample volumes, the carrier frequen-
cies have to be in the GHz range (Shen et
al., 1998).
10.4.4 Diffraction techniques
Probably the most versatile and at the
same time powerful tools for characteriz-
ing phase transformations at high pressure
are diffraction methods. There is a huge
amount of literature on this field. Here we
only briefly present the various domains of
high-pressure diffraction giving appropriate
references for the more interested reader.
10.4.4.1 X-ray diffraction
A comprehensive introduction to the ba-
sic principles of X-ray diffraction is given
by, e.g., Stout and Jensen (1989). X-ray
diffraction at high pressure has initially
only been possible in combination with di-
amond anvil cells. Despite the high trans-
parency of diamonds to X-rays, diffraction
experiments suffer both from the small
sample volume and from shielding and ab-
sorption effects from the pressure cell com-
ponents. This is the reason why high-pres-
sure experiments on sealed-tube sources
are mostly restricted to single crystals, be-
cause the diffraction signal of a crystal of a
size of about 100mm is sufficiently strong
to pass through the diamond anvil (Hazen
and Finger, 1982; Miletich et al., 2000, An-
gel et al., 2000). By measuring accurate
lattice parameters on single crystals (An-
gel, 2000a), we are able to detect and char-
acterize even subtle phase transformations
(Carpenter et al., 1998, Angel, 2000b). In
addition, measurements of single-crystal
diffraction intensities serve to establish an
accurate structural model for the observed
phase transformations and are therefore of
great importance to the understanding of
phase transformations at high pressure.
X-ray powder diffraction is much more
difficult to perform at laboratory sources
(e.g., Haines et al., 1998), mainly because
of the inherently small sample volumes.
For this same reason even at synchrotron
sources the dominating diffraction tech-
nique for investigating powdered samples
has been energy-dispersive diffraction (e.g.,
Holzapfel, 1997), making use of the much
higher flux of a white beam compared
with monochromatic radiation. Energy-dis-
persive diffraction was initially the only
method of choice for in situdiffraction
studies in combination with large-volume
presses (Yagi et al., 1987, Weidner et al.,
1992, Mezouar et al., 1996). In the energy-
dispersive mode, the diffraction signal is
recorded on the energy scale for a fixed
Bragg angle using a Ge solid-state detec-
tor. Modern synchrotron radiation sources,
coupled with new two-dimensional detec-
tor technology, however, allow monochro-
matic powder experiments with both dia-
mond anvil cells (e.g., Fiquet et al., 2000)
and large-volume presses (Parise et al.,
1998, Mezouar et al., 1999a). In this tech-
nique, the diffraction signal of a monochro-
matic X-ray beam is recorded as a function
of the Bragg angle. In contrast to energy-
dispersive data, such angle-dispersive data
allow for a quantitative interpretation of
the diffracted intensities in order to estab-
lish and refine a structural model. Another
advantage of angle-dispersive diffraction
over energy-dispersive techniques is the
higher resolution achievable with mono-
chromatic radiation.
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116. 10.5 Examples 679
Although synchrotron radiation can
overcome the intensity problem caused by
the small sample volumes in high-pressure
powder diffraction experiments, the addi-
tional intensity cannot resolve the problem
of inadequate powder statistics of small
sample volumes. This problem can be
partly alleviated using digital two-dimen-
sional detectors such as online image-plate
detectors or CCD cameras (Shimomura et
al., 1992).
A further step in the use of the unique
properties of synchrotron radiation has re-
cently been made by performing inelastic
X-ray scattering through a diamond anvil
cell in order to determine the acoustic pho-
non dispersion in materials (Krisch et al.,
1997). The application of this very useful
technique in high-pressure research is still
very new and under development.
10.4.4.2 Neutron diffraction
A very comprehensive introduction to
the general principles of neutron diffrac-
tion is given by Bacon (1975). The advan-
tage of neutron diffraction over X-ray dif-
fraction is that neutrons are more sensitive
to light elements. Furthermore, neutron
diffraction is able to distinguish between a
lattice of nuclei and a magnetic lattice.
Neutron powder diffraction at high pres-
sure has been carried out since the 1960s.
The development of the Paris–Edinburgh
press (see Sec. 10.3.2.1) dramatically in-
creased the pressure range accessible for in
situneutron powder diffraction. The quest
towards higher pressures is followed even
more consequently with revolutionary sap-
phire cells (e.g., Goncharenko and Mire-
beau, 1998) allowing for neutron diffrac-
tion experiments up to pressures of 50 GPa.
Another remarkable development in high-
pressure neutron diffraction is the possibil-
ity of performing inelastic single-crystal
neutron diffraction again in combination
with the Paris–Edinburgh cell (Klotz et al.,
1996; 1997). Such data complement the
static picture of a given material as ob-
tained through an elastic scattering experi-
ment with the dynamic nature (phonons) of
the crystalline material.
10.5 Examples
10.5. Zincblende-type semiconductors
Due to their pivotal role in modern elec-
tronic technology, semiconductors are
probably the most intensely studied class
of materials. Experiments at high pressure
are crucial for the testing of theoretical
models, because the application of pressure
is a straightforward way of modifying
the band structure of the material with-
out changing the chemistry. It is therefore
not surprising that semiconductors were
among the very first materials to be inves-
tigated at high pressure (e.g., Bridgman,
1935; Miller and Taylor, 1949). Since then,
theoretical as well as experimental work in
this field has increased greatly.
Application of high pressure on semi-
conductors leads to a closing of the inter-
atomic distances and thus of the gaps
between formerly localized molecular orbi-
tals. Increasing overlap of the orbitals leads
to an increasing delocalization of the elec-
trons, which in turn enhances electrical
conductivity up to a metallization of the
originally insulating or semiconducting
compounds. Narrowing and closing of the
gap between valence and conduction bands
not only has an effect on the electrical con-
duction but also on the optical absorption.
It is for this reason that optical spectros-
copy was one of the first methods used to
study phase transformations in semicon-
ductors at high pressure (Slykhouse andwww.iran-mavad.com
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117. Drickamer, 1958; Paul, 1959; Goñi and
Syassen, 1998).
High-pressure driven phase transforma-
tions have also been extensively studied in
the past 30 to 40 years using diffraction
methods. In the following only an over-
view of phase transformations in zinc-
blende-type semiconductors will be given,
without claiming completeness even in this
limited segment of semiconductors. A most
thorough and comprehensive review of this
subject is given by Nelmes and McMahon
(1998).
First structural diffraction studies on
these materials were performed on pow-
dered samples using energy-dispersive X-
ray diffraction (EDX) (see Sec. 10.4.4).
The picture emerging from these studies is
a sequence of phase transitions where vol-
ume optimization is achieved by a stepwise
increase in coordination number (see Sec.
10.2.2). This increase in coordination num-
ber was believed to happen through the fol-
lowing sequence of phase transformations:
4-fold (diamond, zincblende, wurtzite)
Æ6-fold (NaCl, b-tin) Æ8-fold (simple
hexagonal) Æ8-fold with six close second
nearest neighbors (bcc, CsCl) or 12-fold
(ccp, hcp). For a description of the struc-
ture types relevant for semiconductor crys-
tal chemistry, see Nelmes and McMahon
(1998). The availability of in situangle-
dispersive X-ray powder diffraction
(ADX) (see Sec. 10.4.4) has changed this
simple model dramatically, mainly because
of the much higher resolution of the ADX
technique relative to EDX. The overall pic-
ture emerging from these new experiments
is that at least the change from 4-fold to 6-
fold coordination is characterized by a de-
crease in symmetry, in contrast to the gen-
eral crystal-chemical trend of increasing
symmetry with increasing pressure. The
new 6-coordinated phases can be structu-
rally described as distorted derivatives of
NaCl (Cmcm, cinnabar, Imm2, Immm) or,
b-tin (Imma, simple hexagonal) structures.
An interesting aspect also is that a site-or-
dered version of the diatomic b-tin struc-
ture, which previously was believed to be
of great importance in semiconductor crys-
tal-chemistry, probably does not exist for
any of the group III–V and II–VI semicon-
ductors. A further peculiarity, which has
been found in Si, Ge and GaAs, is the oc-
currence of different phases at a given
pressure, depending whether one is on a
compressional or decompressional branch
of the P-path. In a similar way, InSb, HgSe
and HgTe exhibit intermediate phases dur-
ing their sluggish phase transitions from 4-
fold to 6-fold coordination. These interme-
diate phases (“hidden” phases) cannot so
far be isolated as single phases, but always
occur together with the stable low- or high-
pressure modification. Most of these pecu-
liarities are believed to be linked to site-
ordering problems (non-existence of dia-
tomic b-tin) or to problems of achieving a
true equilibrium state when compressing at
room temperature („hidden“ phases).
Systematic experiments at simultaneously
high temperature and high pressure are
therefore the next experimental step to be
taken to obtain a complete understanding
of semiconductor phase transformations.
In the following the phase transforma-
tions in Si, Ge, GaAs and InSb will be dis-
cussed in more detail, because they are in-
structive examples of most of the special
features encountered in the high-pressure
behavior of group IV, III–V, and II–VI,
semiconductors.
10.5.1.1 Si and Ge
The phase transformations of Si and Ge
are quite similar. The main differences are
found in large discrepancies in the respec-
tive transformation pressures and in an ad-
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118. 10.5 Examples 681
ditional ‘intermediate phase’ in Si at high
pressure. Both Si and Ge crystallize in the
diamond structure at ambient conditions.
The diamond structure can be viewed as
the monatomic version of the zincblende
structure. The zincblende structure is a cu-
bic closest packing (ccp) of anions with
half of its tetrahedral interstices occupied
by the cations. The hexagonal closest pack-
ing (hcp) analog to the zincblende structure
is the wurtzite structure. At 11 GPa (Si)
and 10.6 GPa (Ge), the diamond-type
structure transforms to the b-tin structure.
The exact transition pressure is known to
depend on the amount of stress in the sam-
ple. For silicon, the b-tin structure trans-
forms into an orthorhombically distorted
version of the b-tin structure at 13.2 GPa
(McMahon and Nelmes, 1993). The corre-
sponding phase transformation also occurs
in Ge, but at around 75 GPa (Nelmes et al.,
1996). These orthorhombic phases trans-
form to the previously detected simple hex-
agonal structure at 15.6 GPa (Si) and
~85 GPa (Ge) (Ruoff and Li, 1995; Vohra
et al., 1986). A further increase in pressure
causes Si to adopt another orthorhombic
(Cmca) structure (Si-VI) around 37.6 GPa
(Duclos et al., 1990; Hanfland et al., 1999),
where the Si atoms are 10- and 11-coordi-
nated. A remarkable detail of the Si-VI
phase is that it is iso-structural to Cs-V
(Schwarz et al., 1998). Not only is the to-
pology between these two phases identical,
but also the axial ratios and free atomic co-
ordinates have almost exactly the same val-
ues. Until now, an analogous phase has not
been found for Ge. Instead, Ge is believed
to transform to a hexagonal closest packed
structure with a doubled c-axis (dhcp)
(Vohra et al., 1986) around 100 GPa. The
experimental evidence presented in favor
of such a dhcp structure is, however, scarce
and further experiments are required to
confirm this phase. Si, in turn, is observed
to transform to a hexagonal closest packed
structure (hcp) (Si-VII) around 41.8 GPa
(Duclos et al., 1990) and finally to a cubic
closest packed structure (ccp) around
78 GPa (Duclos et al., 1987). In hcp as well
as ccp, the individual atoms are bonded to
12 nearest neighbors. The ccp structure is
believed to remain stable until at least 248
GPa (Duclos et al., 1990).
On decompression, five more phases
have been found for Si. Slow pressure re-
lease from Si-II (b-tin structure) leads to
the Si-III phase (Besson et al., 1987). It
adopts a body centered cubic structure
(BC8) with 16 atoms per unit cell. The
transition from Si-II to Si-III has recently
been shown to occur via an intermediate
rhombohedral phase (Crain et al., 1994),
which represents a slight rhombohedral
distortion (R8) of the BC8 structure type.
Transition pressures from b-tin to R8 and
R8 to BC8 are 10.1 GPa and 2 GPa, respec-
tively. The decompressional behavior of Si
is different when the rate of pressure re-
lease is very fast. In this case, Si-II (b-tin)
seems to transform into two tetragonal
phases, whose structures are not yet
known.
Slow decompression of Ge-II (b-tin
structure) leads to a phase (Ge-III) with a
non-centrosymmetric, tetragonal structure
(ST12) with space-group P4
32
12. Unex-
pectedly, the BC8 structure, which in Si is
obtained through slow pressure release,
can be grown in Ge by fast decompression
from the b-tin structure.
10.5.1.2 GaAs
GaAs is the III–V analogue of Ge. It is
of very great technological importance,
which explains the large amount of re-
search performed on this compound. De-
spite the intense scrutiny devoted to GaAs,
its high-pressure phase transformations re-www.iran-mavad.com
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119. mained a controversial topic for many
years. GaAs has the zincblende structure at
ambient conditions. A high-pressure phase
transformation to a GaAs-II phase was first
found through electrical measurements
(Minomura and Drickamer, 1962). In the
1980s, it was recognized that the structure
would most probably be of orthorhombic
symmetry. However, space group as well as
structural topology remained ambiguous
(Shimomura et al., 1980; Baublitz and Ru-
off, 1982; Weir et al., 1989) in that a pro-
posed model (space group Pm2m) implied
very close Ga–Ga and As–As contacts.
ADX experiments again proved to be the
key to resolving the puzzle, although the
very similar X-ray scattering power of Ga
and As created additional problems. The
high resolution of the ADX patterns were,
however, able to reveal small reflections,
which allowed for an experimental distinc-
tion between Pm2m and Cmcm (which can
be described as a distorted NaCl structure).
Ga and As form slightly distorted NaCl-
type planes parallel (001). Neighboring
planes are shifted parallel to [010] by about
half a Ga–As distance (Nelmes and McMa-
hon, 1998). Upon further compression,
GaAs has been reported to undergo another
phase transition to a simple hexagonal
structure (Weir et al., 1989). However, a di-
atomic compound such as GaAs, cannot
adopt the simple hexagonal structure, al-
though the topology of the atoms does have
this symmetry. To adopt this structure,
GaAs would have to be completely site-
disordered. It can be speculated that the
similar scattering power of Ga and As may
make a diffraction pattern (because of lim-
ited resolution and signal-to-noise ratio)
appear to have simple hexagonal symme-
try, when in truth it remains of orthorhom-
bic symmetry. This would imply that GaAs
remains in Cmcm structure up to the high-
est pressure investigated, i.e., 108 GPa.
Two more phases of GaAs have been
found. The first occurs upon decompres-
sion of the GaAs-II phase. It leads to a 4-
coordinated cinnabar structure (McMahon
and Nelmes 1997). The other can be grown
by heating the GaAs-II phase to about
450 K at a pressure of ~14 GPa (McMahon
et al., 1998). Its structure (SC16) is the di-
atomic equivalent of the BC8 structure
found in Ge and Si.
10.5.1.3 InSb
InSb, which at ambient conditions also
adopts the zincblende structure, is probably
the most impressive example of a phase di-
agram that has been completely rewritten
in the past 5–10 years. The problem of
very similar scattering powers is even more
severe in InSb than in GaAs. In order to un-
ravel the crystal chemistry of its high-pres-
sure phases, the ADX technique had to be
supported by anomalous scattering in order
to distinguish between true symmetries and
pseudo-symmetries caused by the similar
scattering powers (Nelmes and McMahon,
1998). The peculiar feature of InSb is that
two different successions of high-pressure
phases as a function pressure can be ob-
served, albeit in a reproducible way. The
basic succession is from the zincblende
type to a superstructure of Cmcm (InSb-
IV) around 3 GPa. At 9 GPa the structure
transforms to an orthorhombic distorted
NaCl phase (InSb-III) with space group
Immm. The Immm phase appears to be at-
tained through an intermediate site-disor-
dered orthorhombic phase with space
group Imma. The transformation from
Imma occurs with time (a few hours) or
upon slight heating. InSb-III, in turn, trans-
forms at 17 GPa to a new phase with yet
unknown structure, and upon further com-
pression over 21 GPa, to a site-disordered
body centered cubic structure. A new fea-
682 10 High Pressure Phase Transformationswww.iran-mavad.com
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120. 10.5 Examples 683
ture in this case is that the transition from
the zincblende-type phase to InSb-IV can
also be preceded by an additional phase
transition around 2.1 GPa, which produces
a mixture of a site-disordered b-tin phase
and the InSb-II phase with orthorhombic
symmetry (Immm). This mixture recrystal-
lizes to InSb-IV when left for a few hours.
Even more peculiar is the fact that if, dur-
ing the presence of the mixed phase the
pressure is increased to about 3 GPa within
a few hours, the formation of InSb-IV is
suppressed (Nelmes et al., 1993a). The
InSb-II phase persists up to 9 GPa where it
does not show any phase transition to InSb-
III, because InSb-III already has the Immm
structure.
As outlined in this brief summary of the
unusual high-pressure behavior of GaAs
and InSb, there seems to be a number of
unusual phenomena in the structural be-
havior of zincblende-type semiconductors
at high pressure. These phenomena may
well be linked to kinetic problems as they
are often encountered during phase trans-
formations at high pressure. As shown by
Leinenweber (1993) and Kunz et al.
(1996), such kinetic problems can be alle-
viated by adding high temperature to high
pressure. Very often heating a compressed
sample may eventually lead to a stable
phase. In a similar way, the nonexistence of
a site-ordered b -tin phase could be linked
to microstrain and/or kinetics. Mezouar et
al. (1999b) showed that different phases
can be obtained at a given pressure, de-
pending on the microstrain and hydrosta-
ticity present in a pressure cell. It will thus
be a challenge in future high-pressure semi-
conductor research to combine high pres-
sure and high temperature while simultane-
ously probing the sample.
10.5.2 Materials in the B–C–N system
Diamond (i.e., the high-pressure poly-
morph of carbon) is not only a precious and
popular gem stone, it also has great techno-
logical significance owing to its extraordi-
nary physical properties. The most promi-
nent of these is a Mohs hardness of 10 (di-
amond is the hardest material known) mak-
ing it a most efficient abrasive material.
The reason for the extreme hardness of di-
amond lies in the combination of high co-
valency together with the smallness of the
core electron shell of carbon, distinguish-
ing it from the isomorphic compounds Si
and Ge (see Sec. 10.5.1.1). Also unusual is
the combination of a very high thermal
conductivity coupled with a low electrical
conductivity, which makes it an interesting
material as a heat sink in microelectronic
applications. Graphite (i.e., the low-pres-
sure polymorph of C) also has technologi-
cal importance thanks to its electrical (e.g.,
graphite electrodes) as well as its mechani-
cal (steel additive, graphite composite
material, nanotubes) and lubricating prop-
erties. Therefore, (pseudo-)isoelectronic
systems of C have attracted much attention
in the hope of obtaining materials with
graphite- and diamond-like properties.
There is reason to believe that ternary com-
pounds in the B–C–N system with graph-
ite-like structures might be semiconductors
with high thermal stability or, alternatively,
diamond-type phases in the B–C–N field
may exhibit abrasive properties exceeding
those of diamond combined again with a
much higher thermal stability (Kurdyumov
and Solozhenko, 1999).
Crystallographic investigations of super-
hard materials in situat high pressure pose
an additional challenge: the extreme hard-
ness of the material causes strong devi-
atoric stresses when compressing a pow-
dered sample (Weidner et al., 1994a, b).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

121. The deviatoric strain caused by the devi-
atoric stress leads to anisotropic peak
broadening, which in turn makes the accu-
rate interpretation of powder patterns more
difficult.
10.5.2.1 C
Carbon crystallizes in three different
polymorphs (Fig. 10-9 a). Graphite, the
stable phase at ambient conditions, is char-
acterized by a stacking of hexagonal
layers, where the individual layers are
formed by a honeycomb net of 6-mem-
bered rings formed by C. The layered
structure is responsible for the very low
hardness (Mohs 1–2) as well as the aniso-
tropic electrical properties of graphite.
Graphite can exist in a hexagonal (2H) and
rhombohedral (3R) modification, where
these polytypes differ in the stacking of the
layers: ABABAB for 2H and ABCAB-
CABC for 3R. If graphite is pressurized, it
transforms to diamond. For kinetic reasons,
a pressure of about 8 GPa (5 GPa if a C-so-
lution in a metal melt is used as precursor)
usually has to be combined with a tempera-
ture of ~ 1000 K in order to obtain cubic di-
amond. The crystal structure of cubic dia-
mond is the monatomic equivalent of the
zincblende structure [see Sec. 10.5.1 and
Nelmes and McMahon (1998) for detailed
descriptions of these structure types]. Al-
though the phase boundary has a positive
slope and thus would extend to much lower
pressures at room temperature, graphite
compressed at room temperature does not
readily adopt the diamond structure. In-
stead, pure compression of graphite at
room temperature up to 12 GPa and subse-
quent annealing at about 1000 K leads to
the formation of a hexagonal form of dia-
mond (londsdaleite). This transformation
is reversible in the temperature range
between 1000 and 1300 K. If heated above
1400 K, however, londsdaleite converts
to cubic diamond. Subsequent further
changes in the P–Tfield do not induce any
684 10 High Pressure Phase Transformations
Figure 10-9.P–Tphase diagrams for (a) carbon and (b) BN. Note that the two topologies are identical. (b) is
different from (a) in that it is shifted to lower pressures. This causes the phase boundary between h-BN (graph-
ite-type) and c-BN (diamond-equivalent) to cross the zero pressure line at ~1600 K, thus making c-BN the
stable phase at ambient conditions. After Bundy (1989) and Solozhenko (1994, 1999).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

122. 10.5 Examples 685
further phase transformations as long as we
remain above the graphite–diamond phase
boundary. This implies that londsdaleite,
which crystallizes in the wurtzite structure,
is metastable with respect to diamond. The
stability of diamond under high pressure
seems to be much higher than that of sili-
con or germanium (see Sec. 10.5.1.1). No
phase changes have been observed up to
pressures of 550 GPa (Xu et al., 1986).
This is attributed to the fact that C, unlike
Si and Ge, does not have any core p-elec-
trons, which brings the valence electrons in
C much closer to the nucleus and thus sta-
bilizes the cubic high symmetry structure
(Bundy, 1989).
In the B–C–N phase field, there are four
important ternary phases known, namely
BCN, BC
2N, BC
3N and BC
4N. In addition
to these, there are numerous binary phases
such as BN and several B-rich B–N solid
solutions, B
4C and BC
3along the B–C line
and C
3N
4as the only binary carbon–ni-
tride. Many of these phases can be synthe-
sized as graphite-like compounds through
sputter deposition or chemical vapor depo-
sition (CVD), yielding thin films suitable
for abrasive protection. Because in this
chapter we are mainly interested in high-
pressure phase transformations, we will fo-
cus on structural transformations relevant
to the synthesis of bulk samples.
10.5.2.2 B–N
The III-V analogue to carbon is BN. Ow-
ing to its industrial importance as a dia-
mond substitute in drilling and polishing
applications, a large amount of research
has been done in this field. The topology of
the phase diagram is very similar to that of
carbon (Fig. 10-9). However, it is shifted
towards lower temperatures so that the
phase boundary between the graphite-type
hexagonal phase (h-BN) and a zincblende-
type cubic phase (c-BN) crosses the 1 bar
line at a temperature of about 1600 K. This
makes the zincblende-type c-BN the stable
phase at ambient conditions (Solozhenko,
1994; Solozhenko et al., 1999). However,
the growth of c-BN at atmospheric pres-
sure is only possible in the presence of
supercritical fluids and seed crystals. Spon-
taneous crystallization of c-BN has not
been observed at pressures below 2 GPa
(Solozhenko, 1994). This fact has, of course,
important technological consequences, as
c-BN is the phase with the desired abrasive
properties. The graphite-like phase (h-BN)
is the stable phase at low pressure and high
temperatures. The graphite-like phase can
occur as hexagonal (2H) or rhombohedral
(3R) polytype (e.g., Britun et al., 1999).
Like most binary and ternary phases in the
B–C–N system, h-BN has a turbostratic
structure. This structure can be described
as a graphite structure with good in-plane
ordering but random orientation of the in-
dividual planes around the layer normal.
This leads to the absence of hklor h0l
reflections, while hk0 and 00l reflections
exhibit very irregular peak shapes (e.g.,
Andreev and Lundström, 1994). The phase
boundary between c-BN and h-BN has a
positive slope and meets the liquidus in a
triple point at 3480 K and 5.9 GPa (Solo-
zhenko et al., 1999). Similar to the carbon
phase diagram, the BN also forms a meta-
stable phase (w-BN) at high pressure,
which crystallizes in the wurtzite structure.
The transitions between the various poly-
types are either martensitic or diffusional,
depending on the synthesis and formation
conditions (shock compression vs catalytic
synthesis) as well as on the ordering degree
of the precursor material (Kurdyumov,
1995). A curious but potentially very inter-
esting BN phase has recently been synthe-
sized by laser heating a c-BN sample up to
5000 K at pressures between 5–15 GPawww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

123. (Golberg et al., 1996). This procedure
yielded multi-walled nanotubes, which did
not contain any other inclusions as they are
frequently obtained when producing nano-
tubes via a plasma-arc discharge method.
10.5.2.3 C–N
A lot of effort has been put into the syn-
thesis of sp
3
-bonded carbon nitrides, again
in the hope of creating super-hard materi-
als. This hope has been supported by theo-
retical calculations predicting the stability
of C
3N
4phases whose bulk moduli were
calculated to exceed the value of diamond
(Sung and Sung, 1996; Liu and Cohen,
1989, 1990). These theoretical studies pre-
dict five different phases of C
3N
4. Besides
a graphite-like phase, an a- and a b-phase
are expected. These would be structurally
identical to the corresponding phases of
Si
3N
4. Their expected bulk moduli are 425
GPa and 451 GPa, respectively. A more re-
cent theoretical study (Teter and Hemley,
1996) predicts two more high-pressure
phases. A cubic phase should be structu-
rally identical to the high-pressure Zn
2SiO
4
polymorph. The bulk modulus for this
phase is calculated to be higher than that of
diamond (496 GPa vs. 444 GPa). Nguyen
et al. (1998) managed to synthesize a bulk
C
3N
4phase by laser heating a C
60film with
liquid nitrogen in a diamond anvil cell at
2000 K and 18 GPa. The diffraction pattern
of the retrieved phase is very close to the
one predicted theoretically for the cubic
structure by Teter and Hemley (1996). The
measured compressibility, however, is
much higher than predicted (249 GPa vs.
449 GPa). The phase was found to decom-
pose upon pressure release below 14 GPa.
Successful attempts at bulk synthesis of
C
3N
4have also been reported by He et al.
(1998) and Dymont et al. (1999). In both
cases a precursor containing C, N and H
was pressurized to about 7 GPa and tem-
peratures between 600 and 1200 K. This
procedure yielded faceted crystals of a-
and b-C
3N
4.
10.5.2.4 B–C
The crystal chemistry along the B–C
axis has been well studied (Thévenot,
1990). The most important binary is B
4C.
Its structure is a stuffed derivative of a-
boron, where in B
4C the individual boron
icosahedral chains are linked through a
three-atom C–B–C chain. So far no high-
pressure phases of B
4C are known. How-
ever, owing to the electron-deficient nature
of the intra-icosahedral bonding, boron-
rich solids such as B
4C are expected to be
inverted-molecular solids, where the com-
pressibility of the molecules is higher than
that of the intermolecular links. This is un-
usual because in most cases the compress-
ibility of a molecular crystal is controlled
by the stiffness of the weak van der Waals
bonding between the molecules, while the
bonds within the molecules are rather rigid.
The inverted compressibility was experi-
mentally confirmed for B
4C (Nelmes et al.,
1995). Such an inverted behavior is re-
markable for a compound like B
4C in view
of the very strong and covalent B bonds
within the B
12molecules.
10.5.2.5 B–C–N
Ternary phases in the B–C–N system are
rather difficult to grow, mainly because of
the high stability of BN–C mixtures rela-
tive to true B–C–N phases. Nevertheless,
several ternary phases have been synthe-
sized (see Solozhenko (1997) for a review).
Graphite-type phases with ternary compo-
sitions are mostly grown using CVD meth-
ods or nitration of B- and N-containing
compounds in the condensed state. Investi-
686 10 High Pressure Phase Transformationswww.iran-mavad.com
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124. 10.5 Examples 687
gations of possible diamond-type struc-
tures in this system are difficult owing to
the difficulty of distinguishing true ternary
phases from a mechanical mixture of C and
BN with diffraction methods. In order to
create diamond-type ternary phases (exhib-
iting zincblende- or wurtzite-related struc-
tures), the diffusion during the high-pres-
sure/high-temperature experiment has to be
suppressed, otherwise the ternary phases
decompose into c-BN and diamond. This
can be understood on the basis of the sig-
nificant positive deviation of C
x(BN)
1–x
solid solutions from Vegard’s law. It im-
plies that ternary high-pressure phases will
have a higher volume than a linear combi-
nation of the components involved and will
thus, provided sufficient atomic diffusion
occurs, be unstable at high pressure. The
most efficient way of preventing diffusion
and thus segregation is shock compres-
sion (e.g., Komatsu et al., 1996). Another
possibility of transforming graphite-type
C–B–N material into a diamond-type phase
is by laser heating a sample in a diamond
anvil cell (Knittle et al., 1995). The sam-
ples were grown at pressures between 30
and 50 GPa combined with temperatures
between 2000 and 2500 K. This approach
appears so far to be the only published ex-
ample where a ternary phase could actually
be grown from both ternary graphitic pre-
cursors and mechanical mixtures. It should
be mentioned, however, that the analytical
data of Knittel et al., (1995) could equally
well be interpreted as resulting from me-
chanical mixtures of dispersed diamond
and c-BN (Solozhenko, 1997).
10.5.3 H
2O
Water not only occurs as the life-sustain-
ing liquid phase, but also has an exception-
ally rich phase diagram (Fig. 10-10) in its
solid state. All solid phases of water are
commonly called ice. Presently, there are
12 crystalline and two amorphous phases
of ice accepted. Crystalline ice phases are
labeled with Roman numerals from I to XII
in the approximate sequence of their dis-
covery. The high structural variability can
be explained by a subtle interplay between
enthalpic and entropic contributions. The
flexible O–H…O hydrogen bond with a
strong O–H bond and at least one weak
O…H hydrogen bond around each hydro-
gen enables the formation of a large num-
ber of energetically similar topologies. In
addition, the total energy of the system can
be significantly altered through the entropy
of a disordered distribution of the O–H…O
asymmetry in the framework.
The flexibility of the hydrogen bonds is
intimately linked to their asymmetric
geometry. With simple bond valence and
ionic radius arguments, it is possible to
show that this asymmetry is a direct conse-
quence of the fact that the closest un-
strained O–O distance is longer than the
sum of two relaxed O–H bonds. This would
lead to an underbonding of the hydrogen
atom. As outlined above (see Sec. 10.2.2),
such an underbonding can be compensated
for by an asymmetric coordination geome-
try. This model implies straight O–H..H
geometries for strong hydrogen bonds, but
bent (because under-constrained) O–H…O
configurations for weaker hydrogen bonds
(Brown, 1976, 1995). Therefore the model
correctly predicts the counter-intuitive fact
that high-pressure phases of ice will tend to
have weaker hydrogen bonds. This is due
to the requirement for a denser packing of
water molecules in high-pressure phases
(see Sec. 10.2.1), leading to bent and thus
weakened hydrogen bonds.
The hydrogen bond is not only critical
for H
2O with respect to its structural as-
pects, but also with respect to its electronic
properties. The disordering of the hydro-www.iran-mavad.com
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125. gen bonds strongly increases the polariz-
ability and electrical conductivity of H
2O.
There exists a vast literature on the struc-
ture, physics and chemistry of H
2O, includ-
ing a very comprehensive and thorough re-
view by Petrenko and Whitworth (1999).
A common feature of all ice phases is
that the H
2O molecules form a three-di-
mensional tetrahedral framework. This,
combined with the versatility of the hydro-
gen bonds, makes H
2O an interesting study
case for understanding the possible mecha-
nisms of a framework to reduce its volume.
At low pressures, H
2O adopts rather open
frameworks (see Sec. 10.5.3.1). At pres-
sures above ca. 0.6 GPa, however, it is able
to reduce the molar volume significantly
by forming two interpenetrating, but un-
connected frameworks (see Sec. 10.5.3.5).
In the intermediate range between 0.2 and
0.6 GPa, it adjusts to the imposed volume
constraint mainly by straining the individ-
ual bonds (Sec. 10.5.3.3 and Sec. 10.5.3.4),
which can lead to a weakening of O…H
hydrogen bonds with increasing pressure,
as mentioned above. Under certain condi-
tions, lack of kinetics favors a chaotic
framework collapse leading to an amor-
phous phase (Sec. 10.5.3.6).
10.5.3.1 Ice Ih, XI and Ic
The ice phase occurring at ambient pres-
sure (ice Ih) has a hexagonal structure,
where the oxygen atoms occupy the posi-
tions of the monatomic version of the wurt-
zite structure (see Sec. 10.5.1.1). The hy-
drogen atoms are placed asymmetrically
between neighboring oxygen atoms. Each
oxygen thus forms two O–H bonds and two
O…H hydrogen bonds. The distribution of
the O–H bonds and O…H hydrogen bonds
is disordered, resulting in space group
P6
3/mmc (hcp space group). The local dis-
tribution of the H atoms, however, has to
follow the two ice rules (Bernal and
Fowler, 1933), which are valid for all other
ice phases; (1) there are always two O–H
bonds for each oxygen, (2) there is only
one H per O–O contact. These rules follow
directly out of bond-valence rules and cat-
ion–cation repulsion considerations.
688 10 High Pressure Phase Transformations
Figure 10-10.(a) P–Tphase diagram for H
2O, (b) enlargement of the central section of (a). Note that in (a)
the pressure scale is logarithmic. (After Petrenko and Whitworth, 1999.)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

126. 10.5 Examples 689
The cubic analog to Ih exists as a meta-
stable polymorph designated Ic (ccp space
group Fd3m). It is structurally character-
ized by a diamond-type sub-lattice of the
oxygen atoms. Ice Ic can be formed by re-
covering any of the high-pressure phases in
liquid nitrogen and subsequently heating it
to 120–170 K. There is no way of produc-
ing Ic directly from Ih, heating Ic to about
200 K on the other hand transforms it to
Ih. The energetic difference between Ic
and Ih is only between 13 and 50 Jmol
–1
(Handa et al., 1988). Hexagonal ice can in
principle be transformed into an ordered
polymorph around 72 K (ice XI). However,
the process of collectively rearranging
O-H…H bonds can only take place if there
are a sufficient number of protonic point
defects present, which enable the propaga-
tion of the bond reorientations. Such de-
fects can easily be incorporated through
doping with alkali hydroxides. The phase
transition IhÆXI was first observed
through dielectric permittivity (Kawada,
1972). The exact structure is now agreed to
have the orthorhombic space group Cmc2
1,
with all OH-dipoles parallel c oriented in
a ferroelectric way (Line and Whitworth,
1996; Jackson et al. 1997).
A special and well known common prop-
erty of all these low-pressure phases of
H
2O is that their density (0.92–0.934 g
cm
–3
) is somewhat lower than that of liquid
water.
10.5.3.2 Ice II
The first true high-pressure phase was
discovered in 1900 by Tammann. It can be
formed through compression of Ih at tem-
peratures between –35° and –70°C. Its
structure is quite peculiar in two ways.
First, it has a perfectly ordered arrange-
ment of the hydrogen bond pattern. Sec-
ondly, despite the high pressure of its
stability-field, its structural topology is
characterized by puckered 6-membered
rings of H
2O, which are stacked along the
hexagonal c-axis to form open channels,
similar to the ones formed by Ih. The space
group of ice II is R3
–
. The higher density of
ice II relative to ice Ih is mainly achieved
by the puckering of the 6-membered rings
and the denser stacking of the rings by
means of bending and thus weakening of
the O–H…H bonds. Furthermore, the chan-
nels of ice II can be occupied by He atoms,
creating a He hydrate similar to clathrate
hydrates (Londono et al., 1993). Incorpora-
tion of He actually extends the stability
field of ice II up to the liquidus. Heating ice
II in the absence of He, however, trans-
forms ice II to ice III.
10.5.3.3 Ice III and Ice IX
Ice III is structurally characterized by
puckered 5-membered rings that are mutu-
ally linked in three dimensions to give a
complicated framework of tetragonal sym-
metry, space group P4
12
12 (Kamb and Pra-
kash, 1968). The hydrogen distribution
within the stability field of ice III is highly
disordered (Whalley et al., 1968). How-
ever, it can be ordered by cooling at con-
stant pressure. If the temperature is re-
duced into the stability field of ice II, ice
III remains metastable. Between –65 and
–108°C, the ordering increases gradually,
inducing the phase transformation to ice IX
(the ordered variety of ice III) as revealed
by measurements of electrical permittivity.
Neutron diffraction, however, reveals a sig-
nificant amount of residual ordering also in
ice III (Londono et al., 1993, Kuhs et al.,
1998). Ice III is the high-pressure poly-
morph of H
2O with the lowest density
(1.165 g cm
–3
). However, unlike Ice Ih, its
density at the liquidus is higher than the
corresponding density of the liquid phase,www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

127. resulting in a positive slope of the liquidus
in P–Tspace.
10.5.3.4 Ice V, Ice IV and Ice XII
Further compression of ice III or ice II
leads to ice V. The structure of ice V is a
complex framework built by 4- and 5-
membered oxygen rings (Kamb et al.,
1967). Each oxygen atom is tetrahedrally
surrounded by four neighboring oxygen at-
oms. The symmetry is monoclinic with
space group A2/a. The hydrogen atoms are
distributed in a disordered arrangement
between the oxygen links. In fact, the two-
fold axis of this space group does not allow
for any proton ordering. Proton ordering, if
it occurs, would thus imply a noncentro-
symmetric space group Aa.
A special feature of ice V is that in its
stability field, at least two ice phases with-
out their own stability fields have been de-
scribed. Ice IV has been observed within
the stability field of ice II, ice V and ice VI.
Even a liquidus for ice IV, which lies
within the field of these solid phases has
been established (Chou and Haselton,
1998). The structure of ice IV is quite re-
markable. Only slightly puckered 6-mem-
bered rings are linked together within a
plane by strongly elongated 6-membered
rings forming a puckered plane with
trigonal symmetry. The planes are stacked
in an ABABAB sequence where A- and B-
planes are shifted relative to each other by
the value of the hexagonal a-axis. Like-
type planes (A–A and B–B, respectively)
are linked together by hydrogen bonds
through the regular 6-membered rings of
the neighboring planes. This results in a
single, self-penetrating framework with a
higher density (1.436 g cm
–3
) than ice V
(1.402 g cm
–3
).
Only recently, another new phase of ice
has been described by Lobban et al. (1998)
within the stability field of ice V. The
new phase can reproducibly be grown at
0.55 GPa and 260 K with slow cooling
rates. Its structure consists of two symmet-
rically independent sets of oxygen atoms.
The first one forms zig-zag chains parallel
c. Neighboring chains are rotated by 90°.
These chains are connected by the sec-
ond type of oxygen atoms thus forming
a tetrahedral framework, whose density
(1.437 g cm
–3
) is very close to that of ice
IV. While in ice IV, the increase in density
relative to ice V is achieved by self-pene-
tration, ice XII compresses by additional
bending. In both metastable phases, the hy-
drogen positions are disordered between
the O–O links.
10.5.3.5 Ice VI, Ice VII, Ice VIII, Ice X
Compressing ice V beyond 0.6 GPa in-
duces a phase transformation to ice VI.
This is the first example of an ice phase,
where the structure is characterized by two
interpenetrating frameworks which are not
linked together (‘self-clathrates’, Kamb,
(1965)). In ice VI, the topologies of two
frameworks are identical. They are sym-
metrically related by an inversion and
shifted relative to each other by a/2 + b/2 +
c/2. The individual frameworks can be de-
scribed as chains of tetrahedra, extended
along c. All tetrahedra of one framework
have the same orientation. The chains of
each framework are connected parallel a
and bwith each other, but unconnected
with the chains of the other sublattice.
Ice VII has, together with ice VIII and
ice X, the densest H
2O topology. Its pack-
ing is as efficient as it is simple and can be
described by two interpenetrating frame-
works of the cubic ice Ic. The hydrogen po-
sitions in ice VII are completely disordered
(Walrafen et al., 1982; Kuhs et al., 1984).
The interesting aspect of this structure is
690 10 High Pressure Phase Transformationswww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

128. 10.5 Examples 691
that the oxygen atoms form a body cen-
tered cubic lattice. This implies that each
oxygen atom has eight nearest neighbors.
However, it forms hydrogen bonds only to
four of them in tetrahedral orientation and
in this way forms two interpenetrating but
independent frameworks. This also implies
that the O–O distances to the hydrogen-
bonded oxygen neighbors are equivalent to
the non-bonded O–O contacts.
The disorder–order transition from ice
VII to ice VIII was first observed through
dielectric measurements (Whalley et al.,
1966). Although the structure crystallizes
in the tetragonal space group I4
1/amd, the
positions of the oxygen atoms deviate only
little from a body centered cubic lattice,
confirming the order–disorder nature of the
phase transformation of ice VII to ice VIII
(Kuhs et al., 1984). The ordering pattern of
the O–H bonds is ferroelectric within one
sub-lattice, but antiferroelectric between the
two frameworks. In ice VIII, the O–D dis-
tances remain almost constant with increas-
ing pressure (Nelmes et al., 1993b), al-
though the O–O distances decrease signifi-
cantly. This leads to a decrease in the dis-
tance between the two potential wells of the
hydrogen position. Therefore, the entropy
difference between the ordered and disor-
dered state decreases gradually with in-
creasing pressure, leading to a progressively
steeper phase boundary VII–VIII, which
eventually ends as a vertical slope at 62 GPa
(Pruzan et al., 1993). At this point, the en-
ergy barrier between the two proton posi-
tions is sufficiently small to be crossed with
the thermal energy of the protons, thus lead-
ing to a dynamic disorder. This phase thus
has the average position of the hydrogen at-
oms in the middle between the two oxygen
atoms. An identical structural configuration
can be expected for a structure with a single
potential minimum as is predicted for ice X
at pressures exceeding 62 GPa.
In ice X, the O–O contacts are squeezed
sufficiently close together that the hydro-
gen can sit unstrained in the center of the
O–O-link. Such a symmetric O–H–O ar-
rangement thus has no potential for disor-
der in the first place. Such a phase is differ-
ent from a double-well phase with a dy-
namically disordered proton, which from a
physical point of view should still be called
ice VII. Ice X has been predicted by Kamb
and Davis (1964). Although experimental
evidence for a symmetric phase was ob-
served from IR spectroscopic studies by
the appearance of a new mode characteris-
tic of a symmetric O–H–O configuration
(Goncharov et al., 1996; Aoki et al., 1996,
Pruzan et al., 1997), it should be pointed
out that up to now there is no direct experi-
mental evidence for a truly proton-ordered
single-well phase of ice X. This lack of ex-
perimental evidence is due to the difficulty
of performing experiments that are able to
localize hydrogen or deuterium positions in
the pressure range of 60 GPa and more.
10.5.3.6 Amorphous ice at high
pressure
As mentioned above, there are two
amorphous high-pressure phases of H
2O in
the solid state. A first phase is formed by
compressing ice Ih at 77 K (where slow ki-
netics inhibit the recrystallization of ice II)
to the extrapolated liquidus of Ih. Its den-
sity in the quenched state is 1.17 g cm
–3
.
Upon heating, this high-density amorphous
(HDA) phase transforms to a second amor-
phous phase with lower density (0.94 g
cm
–3
) at 125 K (LDA). LDA, in turn can be
transformed into HDA at 77 K by compres-
sion to 0.6 GPa.
The research on the high-pressure phases
of ice is far from completed. The under-
standing of the various amorphous phases
and the theoretical and experimental questwww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

129. for ice X and beyond are just the most
prominent examples of a lively topic with
relevance to physical chemistry and mate-
rials science.
10.6 Acknowledgements
The author wishes to thank Vladimir
Solozhenko, Richard Nelmes, Mohammed
Mezouar, Guillaume Fiquet, and Denis An-
drault for valuable discussions. This work
received support from the ‘Schweizerische
Nationalfonds’ (Swiss National Science
Foundation) through grant # 21-052682.97.
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p. 141.www.iran-mavad.com
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133. 2 Solidification
Heiner Müller-Krumbhaar
Institut für Festkörperforschung, Forschungszentrum Jülich, Germany
Wilfried Kurz
Département des Matériaux, EPFL, Lausanne, Switzerland
Efim Brener
Institut für Festkörperforschung, Forschungszentrum Jülich, Germany
List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.1 Introduction................................ 85
2.2 Basic Concepts in First-Order Phase Transitions............ 86
2.2.1 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.2.2 Interface Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.2.3 Growth of Simple Crystal Forms . . . . . . . . . . . . . . . . . . . . . . . 91
2.2.4 Mullins–Sekerka Instability . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.3 Basic Experimental Techniques....................... 95
2.3.1 Free Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.3.2 Directional Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.4 Free Dendritic Growth........................... 100
2.4.1 The Needle Crystal Solution. . . . . . . . . . . . . . . . . . . . . . . . . 101
2.4.2 Side-Branching Dendrites . . . . . . . . . . . . . . . . . . . . . . . . . . 108
2.4.3 Experimental Results on Free Dendritic Growth . . . . . . . . . . . . . . 115
2.5 Directional Solidification.......................... 120
2.5.1 Thermodynamics of Two-Component Systems . . . . . . . . . . . . . . . 121
2.5.2 Scaled Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.5.3 Cellular Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2.5.4 Directional Dendritic Growth . . . . . . . . . . . . . . . . . . . . . . . . 131
2.5.5 The Selection Problem of Primary Cell Spacing . . . . . . . . . . . . . . 135
2.5.6 Experimental Results on Directional Dendritic Growth . . . . . . . . . . . 139
2.5.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
2.6 Eutectic Growth.............................. 152
2.6.1 Basic Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
2.6.2 Experimental Results on Eutectic Growth . . . . . . . . . . . . . . . . . . 158
2.6.3 Other Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
2.7 Summary and Outlook........................... 164
2.8 References.................................. 165
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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134. List of Symbols and Abbreviations
a distance
A area
A amplitude
Aˆ differential operator
a
I,a
K constants
b arbitrary parameter
Bˆ self-adjoint differential operator
DC miscibility gap
C,C¢ constants
C
E eutectic concentration
C
L,C
S,C
a, etc. concentration
c
p specific heat at constant pressure
d atomistic length, capillary length
D diffusion constant
d
0 capillary length
D
i transport coefficients, diffusion constants
D
T thermal diffusion coefficient
E
1(P) exponential integral
erfc error function complement
F Helmholtz energy
f scaling function
g Gibbs energy density
DG change in Gibbs energy
Gˆ Gibbs energy per surface element
G({
x
i}) Gibbs energy
G
T constant temperature gradient
J probability current
K curvature
k segregation coefficient, wave number
k
B Boltzmann constant
L natural scale
l external length, diffusion length
l
˜
thickness of layer, diffusion length
L
m latent heat of melting
m
a,m
b liquidus or solidus slope
N
i particle number of speciesi
P pressure, Péclet number
∫ principal value
P({
x
i}) probability of configuration
q wave number, inverse length
Q
min minimal work
r radius of nucleus, radial distance
R(S) local radius of curvature at pointS
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135. List of Symbols and Abbreviations 83
R
0 radius of curvature
r
c critical radius
S position on the interface, entropy
DS change in entropy
T temperature
t time
Dt change in time
T
0 reference temperature
T
E eutectic temperature
T
I interface temperature
T
m melting temperature
u dimensionless temperature field or concentration field
U energy
V speed
V
a maximal speed, absolute stability
V
c critical velocity
v
R growth rate
w probability of fluctuation
x space coordinates
X
i,X
j extensive variables
Y
i,Y
j intensive variables
Z partition function
z space coordinate
b interface kinetics
b
4 4-fold anisotropy of the kinetic coefficient
g surface tension or surface free energy
G ratio of S–L interface energy to specific melting entropy
G
2 Green’s function integral
D supercooling (negative temperature fielduat infinity)
d local concentration gap, surface tension anisotropy
e relative strength of anisotropy of capillary length
z(x,t) deviation of the interface
h coordinate
J angle of orientation relative to the crystallographic axes
Q function of (x,t)
k(s) mobility
l interface spacing, wavelength
L(s,e) solvability function
l
2 wavelength of side branch
l
f wavelength of the fastest mode
l
s stability length
m chemical potential
n
0 effective kinetic prefactor
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136. x
i position vectors
D
x
i small normal displacement
r Ivantsov radius
s scaling function
t variable
W atomic volume, atomic area
DLA diffusion limited aggregation
DS directional solidification
f faceted
l.h.s. left-hand side
nf non-faceted
PVA pivalic acid
r.h.s. right-hand side
SCN succinonitrate
WKB Wenzel–Kramers–Brillouin technique for singular pertubations
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137. 2.1 Introduction 85
2.1 Introduction
This chapter on solidification discusses
the basic mechanisms of the liquid–solid
phase transformation. In particular, we ad-
dress the phenomena of cellular and den-
dritic patterns formed by the interface be-
tween liquid and solid, as the interface, or
solidification front, advances into the liq-
uid during the solidification process.
The atomistic processes of the liquid–
solid transformation are still not well un-
derstood, so we will use a phenomenologi-
cal level of description. The processes
on very large scales, such as casting or
welding, depend greatly on the experi-
mental equipment and are discussed, for
example, by Flemings (1991) and Mordike
(1991).
We will therefore restrict our attention to
phenomena occurring on some important
intermediate length scales. There is a natu-
ral scaleLof the order of micrometers (or
up to millimeters) that gives a measure for
the microcrystalline structures formed dur-
ing the solidification process. In its sim-
plest form this natural length is the geomet-
ric meanL~
dlof a microscopic intrinsic
lengthddefined by typical material proper-
ties and an external lengthldefined by the
macroscopic arrangement of the experi-
mental equipment. The intrinsic correlation
lengths in liquids and solids near the freez-
ing point are rather short, of the order of
atomic size, or several Ångströms, because
solidification is a phase transition of first
order. By contrast, the experimental equip-
ment gives external length scales in the
range of centimeters to meters, such that
we consequently arrive at the mentioned
scale of micrometers.
Assuming for the moment that only two
lengths are important, why should we ex-
pectLto be given by the geometric mean
rather than, for example, the arithmetic
mean? An intuitive argument goes as fol-
lows: the patterns formed at the solid–liq-
uid interface and in both adjacent phases
during the solidification process result
from the competition of two “forces”, one
being stabilizing for homogeneous struc-
tures, the other being destabilizing. The
stabilizing force here clearly must be asso-
ciated with the intrinsic atomistic lengthd,
since we have argued that it is related to the
length of correlation or coherence inside
the material or at the interface. In contrast,
we must associate the external lengthlwith
a destabilizing force. Again this is a quite
natural assumption, as the phase transfor-
mation or destabilization of the nutrient
phase is induced by the experimental envi-
ronment.
It is now obvious that the result of such a
competition of “forces” should be ex-
pressed by the productdlof the two repre-
sentative quantities rather than by the sum,
since the latter would change the relative
importance of the two lengths when their
values become very different.
Admittedly, these arguments look a little
overstressed considering the many parame-
ters controlling the details of a solidifica-
tion process. Note, however, that nothing
has been said so far about the precise rela-
tion ofd,landLto any specific process,
nor have we defined the proportionality
factor. In principle,dandlcould also enter
with different exponents but fortunately
things are not usually that complicated
right from the start.
If we are still courageous enough to
make one more step on this slightly unsafe
ground, we may finally assume that the ex-
ternal length scalelis related (destabilizing
force!) to the speedVof the solidification,
which gives a length when combined with
a diffusion constantDfor heat or matter as
l~D/V. From this we immediately obtain a
relation between the speed of phase changewww.iran-mavad.com
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138. Vand the length scaleLof the resulting
pattern:
VL
2
≈constant (2-1)
Surprisingly enough, we have arrived at
about the most celebrated equation for pro-
cesses of dendritic, directional and eutectic
solidification, without even defining any of
these processes! Two remarks, therefore,
may be in order here. First, detailed studies
of the different solidification schemes in
recent years have attempted to extract the
common and universal aspects of these
processes. Such common features indicate
a common basis of rather general nature, as
outlined above. Second, we have of course
ignored most of the specific aspects of each
individual process. In directional solidifi-
cation, for example, a band of possible
wavelengths for stationary patterns are
found and up to now it is not clear if and
how a specific wavelength from that band
is finally selected. The assumption of just
two independent length scales in many
cases is also a rather gross simplification of
the actual situation. We will therefore leave
this line of general arguments and look at
some concrete models that are believed to
capture at least some essentials for the fas-
cinating patterns produced during solidifi-
cation.
Some remarkable progress has been
achieved in the theoretical treatment of
these phenomena during the recent years.
In the list of references, we have concen-
trated our attention on recent developments
since there are some good reviews on older
work (for example, Langer, 1980a; Kurz
and Fisher, 1998).
An experimentalist may finally wonder
why we have expressed most of the equa-
tions in a non-dimensionally scaled form
rather than writing all material parameters
down explicitly at each step. One reason is
that the equations then appear much sim-
pler than if we attempted to carry along all
prefactors. The second and more important
reason is that the scaled form allows for a
much simpler comparison of experiments
for different sets of parameters which usu-
ally appear only in some combinations in
the equations, thereby leading to cancella-
tions and compensations.
Section 2.2 gives a quick summary of the
ingredients for a theory, starting at nuclea-
tion, then deriving boundary conditions for
a propagating interface between two
phases, and finally discussing some gen-
eral aspects of the diffusion equation with a
propagating boundary. This is followed by
an introduction to basic experimental tech-
niques in Sec. 2.3. In Sec. 2.4, the case of a
simple solid growing in a supercooled melt
is discussed in some detail, leading to the
present understanding of dendritic growth.
In Sec. 2.5 the technically important pro-
cess of directional solidification is pre-
sented. The evolution of cellular patterns
above a critical growth rate can in principle
be understood without any knowledge of
dendritic growth. Actually, the parameter
range for simple sinusoidal cells is very
narrow so that we usually operate in the
range of deep cells or even dendrites,
which suggests our sequence of presenta-
tion. Finally, these concepts are extended
in Sec. 2.6 to alloys with a eutectic phase
diagram and the resulting complex phe-
nomena. As this field is currently in rapid
theoretical development, our discussion
here necessarily is somewhat preliminary.
The chapter is closed by a summary with
complementary remarks.
2.2 Basic Concepts
in First-Order Phase Transitions
The different possible phases of a mate-
rial existing in thermodynamic equilibrium
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139. 2.2 Basic Concepts in First-Order Phase Transitions 87
must be discriminated by some quantity in
order to formulate a theory. Such quantities
are called “order parameters” and should
correspond to extensive thermodynamic
variables. The difference between a solid
and a liquid is defined by the shear mod-
ulus, which changes discontinuously at the
phase transition. This definition describes
the difference in long-range orientational
correlations between two distant pairs of
neighboring atoms. Normally, we use in-
stead the more restrictive concept of trans-
lational order as expressed through two
point correlation functions, or Bragg peaks,
in scattering experiments.
Although these different order-parame-
ter concepts pose a number of subtle ques-
tions, particularly in two dimensions where
fluctuations are very important, we will
simply assume in this chapter that there
is some quantity which discriminates be-
tween a solid and a liquid in a unique way.
Such an order parameter may be the den-
sity, for example, which usually changes
during melting, or the composition in a
two-component system. We would like to
stress, however, that these are just auxiliary
quantities that change as a consequence of
the solid–liquid transition but which are
not the fundamental order parameters in
the sense of symmetry arguments. For a
more general discussion see the Chapter by
Binder (Binder, 2001) and literature on or-
der–disorder transitions (Brazovsky, 1975;
Nelson, 1983).
First-order transitions are characterized
by a discontinuous change of the order pa-
rameter. All intrinsic length scales are
short, typically of the size of a few atomic
diameters. We may therefore assume local
thermal equilibria with locally well-de-
fined thermodynamic quantities like tem-
perature, etc., and smooth variations in
these quantities over large distances. Inter-
faces in such systems will represent singu-
larities or discontinuities in some of the
quantities, such as the order parameter or
the associated chemical potential, but they
will still leave the temperature as a contin-
uous function of the position in space.
2.2.1 Nucleation
A particular consequence of the well-de-
fined local equilibrium is the existence of
well-defined metastable states, correspond-
ing to a local, but not global, minimum in
the free energy. But so far we have ne-
glected thermal fluctuations. The probabil-
ity of a fluctuation (or deviation from the
average state) of a large closed system is
w~ exp (D S) (2-2)
whereDSis the change in entropy of the
system due to the fluctuation (see, e.g.,
Landau and Lifshitz, 1970). DefiningQ
min
as the minimal work necessary to create
this change in thermodynamic quantities,
we obtain
DS=Q
min/T
0 (2-3)
withT
0being the average temperature of
the system. Note that this holds even for
large fluctuations, as long as the change of
extensive quantities in the fluctuation re-
gion is small compared with the respective
quantities in the total system.
Considering this system as a metastable
liquid within which a fluctuation has
formed a small solid region, and assuming,
furthermore, that the liquid is only slightly
metastable, we arrive at the well-known re-
sult (Landau and Lifshitz, 1970) in three
dimensions for a one-component system:
(2-4)
which together with Eqs. (2-2) and (2-3)
gives the probability for the reversible for-
Q
r
PPr
min [() ()]=
LS−−+
4
3
4
3
2
p
pW
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140. mation of a spherical solid nucleus of ra-
diusrwithin a slightly supercooled liquid.
Here
Wis the atomic volume,Pthe pres-
sure,
m
L>m
Sare the chemical potentials of
liquid and solid in a homogeneous system,
and
gis the (here isotropic) surface ten-
sion, or surface free energy.
A few remarks should be made here.
When deriving Eq. (2-4), we use the con-
cept of small deviations from equilibrium,
while Eqs. (2-2) and (2-3) are more gener-
ally valid (Landau and Lifshitz, 1970). In
the estimation of the range of validity of
Eq. (2-4), it is apparent, however, that it
should be applicable to even very large
supercoolings for most liquids, since the
thermal transport is either independent of,
or faster than, the kinetics of nucleus for-
mation (Ohno et al., 1990). The range of
validity of Eq. (2-4) is then typically lim-
ited by the approach to the “spinodal” re-
gion, where metastable states become un-
stable, even when fluctuations are ignored
(see the Chapter by Binder and Fratzl, 2001).
Assuming, therefore that we are still in
the range of well-defined metastable states,
we may write Eqs. (2-2) and (2-3) as
w=
n
0exp (–DG/T
0) (2-5)
identifying the change in Gibbs energyDG
by Eq. (2-4), with an undetermined prefac-
tor
n
0. Here we do not discriminate
between surface tension and surface free
energy, despite the fact that the first is a
tensorial quantity, and the latter only scalar
(although it may be anisotropic, which is
ignored here). A difference between sur-
face tension and surface free energy arises
when the system does not equilibrate
between surface and bulk so that, for exam-
ple, the number of atoms in the surface
layer is conserved. Throughout this chap-
ter, we will assume perfect local equilibra-
tion in this respect, and we may then ignore
the difference.
The extremal value of Eqs. (2-4) and (2-5)
with respect to the variation ofrgives the
critical radius
(2-6a)
or
(2-6b)
so that forr<r
c, the nucleus tends to
shrink, while forr>r
cit tends to grow and
atr=r
cit is in an unstable equilibrium.
The same thermal fluctuations causing
such a nucleus to appear also produce devi-
ations from the average spherical shape.
This leads to power-law corrections in the
prefactor of Eq. (2-5) or logarithmic cor-
rections in the exponent (Voronkov, 1983;
Langer, 1971).
So far these considerations have dealt
with static aspects only. Since the fluctua-
tions vary locally with time, Eq. (2-5) may
be interpreted as the rate at which such
fluctuation occurs, and, consequently, with
r=r
cwe obtain the rate for the formation
of a critical nucleus which, after appear-
ance, is assumed to grow until the new
phase fills the whole system. This is the
classical nucleation theory. A very elegant
formulation was given by analytic continu-
ation into the complex plane (Langer,
1971). Further additions include the defini-
tion of the prefactor in Eq. (2-5) and a more
detailed analysis of the kinetics nearr=r
c
(Zettlemoyer, 1969, 1976).
Considering the many uncertainties en-
tering from additional sources such as the
range of atomic potentials and the change
of atomic interaction in the surface, we will
ignore all these effects by absorbing them
into the effective kinetic prefactor
n
0in Eq.
(2-5), to be determined experimentally.
r
c= (2-dim)
g
m
D
W
r
c= (3-dim)
2
g
m
D
W
88 2 Solidificationwww.iran-mavad.com
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141. 2.2 Basic Concepts in First-Order Phase Transitions 89
2.2.2 Interface Propagation
An interface between two regions in
space of different order parameters (solid
vs. liquid) will be treated in this chapter as
ajump discontinuityand as an object of in-
finitesimal thickness. In this section we de-
rive a local equation of motion for the inter-
face, which will serve as a boundary condi-
tion in the remaining part of the chapter.
For simplicity, consider a one-dimen-
sional interface in a two-dimensional sys-
tem. Ignoring the atomic structure, assume
that the interface is a smooth continuous
line. Marking points at equal distancesaon
this line, we may then define velocities of
the points in the normal direction as
(2-7)
whereD
x
iis the small normal displace-
ment of pointi. If we assume these points
to be kept at fixed positions
x
ifor the mo-
ment, a (restricted) Gibbs energyG({
x
i})
may be assigned to this restricted interface.
The probability of this configuration is
P({
x
i}) =Z
–1
exp (–G/T) (2-8)
with temperatureTin units ofk
BK andZ
the partition function. Since the total prob-
ability is conserved, we obtain the continu-
ity equation
(2-9)
withJ={J
i} as the probability current in
i-space
(2-10)
and the divergence taken in the same ab-
stract space. The first term in Eq. (2-10) is
a drift, the second term the constitutive re-
lation with transport coefficientsD
i,and
the derivative is taken normal to the inter-
JVPD
P
ii i
i= −
∂
∂
x
∂
∂
+
P
t
idiv =
()J0
V
t
i
i=
D
D
x
face. We now assume local equilibrium to exist on length scalesa(i.e.iÆi±1) such
that the probability current is zero
(2-11)
and with Eqs. (2-7) and (2-8), we immedi- ately arrive at
(2-12)
Taking the continuum limitaÆ0, we ob-
tain the final form
(2-13)
This is the time-dependent Ginzburg–Lan- dau equation (Burkhardt et al., 1977). Here Sdenotes a position on the interface,∂
x(S)
is the normal displacement,d/d
x(S)the
variational derivative, and
k(S) the “mo-
bility”, which may depend on position and orientation.Gˆis the Gibbs energy per sur-
face element.
We will now make some explicit as-
sumptions aboutGin order to arrive at an
explicit equation of motion. Letnˆ
Sbe the
normal direction on the interface,
g(nˆ
S)the
interface free energy,
Jthe angle of orien-
tation relative to the crystallographic axes, R(S) the local radius of curvature at pointS
on the interface, andA
L,Sthe areas cov-
ered by liquid or solid. The Gibbs free en- ergy of the total solid–liquid system with interface is
(2-14)
from which the variational derivative in Eq. (2-13) is formally obtained by
(2-15)
d
d
d
dGS
G
S
S
tot
=d∫
⎡
⎣
⎢
⎤
⎦
⎥
ˆ
()
x
x
GS A
A
S
tot
Liquid
LL
Solid
SS
=d d
d∫∫
∫
+
+g g
g
()
()
x
x
∂
∂
−x
k
x()
()
ˆ
()
S
t
S
G
S
=
d
d
∂
∂
−
∂
∂x
x
ii i
i
t
D
T
G
=
({ })
xx
VP D
P
ii
i=
∂
∂
xwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

142. whered x(S) is a small arbitrary displace-
ment of the interface in the normal direc-
tion. Explicitly this is written as
(2-16)
with Gibbs energy densitiesggiven here
per atomic “volume” (or area)
W. As usual,
we have to extract termsd
xout of the
termsd(dS) andd
g. Assuminggto depend
only on local orientation,
(2-17)
and with
(2-18)
we obtain
(2-19)
The other term simply gives
(2-20)
Incorporating Eqs. (2-19) and (2-20) into
Eq. (2-16) and integrating Eq. (2-19) by
parts, we obtain through comparison with
Eq. (2-15)
(2-21)
as an explicit local equation for the ad-
vancement of an interface in two dimen-
sions with normal velocityV
^, anisotropic
kinetic coefficient
k(J), surface Gibbs en-
ergy
g(J), and jumpDg=g
L–g
Sof Gibbs
energy density at positionSalong the inter-
face (Burkhardt et al., 1977). Here, the
Gibbs energy densitygcorresponds to an
∂
∂
≡−+
⎛
⎝
⎜
⎞
⎠
⎟
⎧
⎨
⎩
⎫
⎬
⎭
⊥
x
kg
g
t
VS
R
=
d
d
() ()
J
JDg
1
2
2
dd()d= dS
R
S
1 x
ddg
g
x()J
J=
d
d
d
d
−
1
RS
ddJ
J
J
J=
d
d
d
d
=
d
d
d
d
d
d
=
SSS S Rx;; −
1
ddg
g()J
J
J=
d
d
ddd
d
GSS
SS S S
S
S
tot
LS
={ d d
d∫
∫ +
−−gg
x() }
()[ () ()]gg
infinite solid or liquid without influences
from curvature terms. For a single-compo-
nent system,
m=gW, wheremis the chemi-
cal potential of the respective phase. The
generalization to three dimensions adds an-
other curvature term into Eq. (2-21), which
then corresponds to two curvatures and an-
gles in the two principal directions (for iso-
tropic
g, the curvature 1/Ris simply
changed to 2/R).
Two useful observations can be made at
this stage. There is a solution withV
^=0if
the term {…} in Eq. (2-21) is zero. For fi-
nite radius of curvatureR, this corresponds
precisely to the critical nucleus, Eq. (2-6b),
but now with anisotropic
g(J). This equa-
tion therefore determines the shape of the
critical nucleus in agreement with theWulff
construction(Wulff, 1901). Second, for
very large mobility
k(J) the deviation
from equilibrium {…} may be very small
in order to produce a desired normal veloc-
ityV
^. We will use this simplification of
equilibrium at the interface
(2-22)
wherever possible, but we will comment on
the changes due to finite
k(J). The double
prime means derivatives as in Eq. (2-21).
In many cases this seems to be justified
by experimental conditions. On the other
hand, very little is known quantitatively
about
k(J). A last point to mention here is
our assumption that the interfaces are not
faceted at equilibrium. If they are faceted,
as crystals typically are at low tempera-
tures in equilibrium with their vapor, the
situation is considerably more complicated
and not completely understood (Kashuba
and Pokrovsky, 1990).
It is now generally believed that faceted
surfaces undergo a kinetic roughening tran-
sition even at small driving forces, so that a
rough surface is present under growth con-
Dg≅+ ′′
1
R
()(gg 2-dim)
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143. 2.2 Basic Concepts in First-Order Phase Transitions 91
ditions. The concept outlined above then
generally applies in an effective fashion.
Actually, facets may persist over experi-
mentally relevant length scales, which are
not covered by this analysis in the kinetic
region. A summary of theoretical develop-
ments can be found in the article by Krug
and Spohn (1992).
To complete this section on the basic
theoretical ingredients, we will now dis-
cuss the influence of diffusive transport of
heat or matter on the propagation of a
solid–liquid interface (see also Langer,
1980a).
2.2.3 Growth of Simple Crystal Forms
a)Flat interface
The simplest model for a solidifying
system consists of two half-spaces filled
with the liquid and solid of a one-compo-
nent material of invariant density and sep-
arated by a flat interface. The interface is
approximately in equilibrium (Eq. (2-22))
at melting temperatureT
mbut advancing at
a speedVin the positivez-direction into the
liquid. During this freezing process, latent
heatL
mhas to be transported into the su-
percooled liquid, the solid remaining atT
m.
The equation of motion is then the ther-
mal diffusion equation
(2-23)
with thermal diffusion coefficientD
Tand
appropriate boundary conditions. At infin-
ity in the supercooled liquid the tempera-
tureT
•<T
mis prescribed, andT
mis the
temperature at the interface. It is now very
convenient to replace the temperature field
Tby a dimensionlessuthrough the trans-
formation
(2-24)
ut
TtT
Lc
p
(,)
(,)
(/)
x
x
=
m
m−
∂
∂
∇
t
TtD Tt
T(,) (,)xx=
2
wherec
pis the specific heat of the liquid at
constant pressure. If, instead of tempera- ture or heat, a second chemical component is diffusing, a similar transformation to the same dimensionless equations can be made. This is described in Section 2.5.2. We would like to stress the importance of such a scaled representation as it allows us to compare at a glance experimental situa- tions with different sets of parameters.
In this dimensionless form, the equation
of motion is
(2-25)
and the boundary conditions are
u=u
•< 0 forzÆ• (2-26a)
u= 0 at interface (2-26b)
So far we have not specified how the inter-
face motion is coupled to the equation of
motion. Obviously, the latent heatL
mgen-
erated during this freezing process at a rate
proportional toVhas to be carried away
through the diffusion field. This requires a
continuity equation at the interface
V=–D
Tnˆ·—u (2-27)
and since in the model here only thez-axis
is important
(2-28)
The growth rate is therefore proportional to
the gradient of the diffusion field at the
interface in the liquid. As the interface is
moving, we conveniently make a coordi-
nate transformation from {z,t}for{z¢,t¢}
into a frame of reference moving along
with the interface atz¢=0:
(2-29)
′−
′∫zz V
tt
t
=d
=
0
()tt
VD
z
uzt
T= at interface−
∂
∂
(,)
∂
∂
∇
t
uD u
T=
2www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

144. Performing that transformation, we obtain
the following for the diffusion equation
(after having again dropped the primes for
convenience):
(2-30)
which defines a diffusion length
l∫2D
T/V (2-31)
The interface is now always atz= 0, which
makes the boundary condition Eq. (2-26b)
and the continuity Eq. (2-28) definite.
We can obtain a stationary solution by
combining Eq. (2-30) with Eqs. (2-26) and
(2-28) when∂u/∂t=0:
(2-32)
This equation is consistent with the as-
sumption thatu=0 (T=T
m) everywhere in
the solid.
This equation describes the diffusion field
ahead of the interface. It varies exponen-
tially from its value at the interface toward
the value far inside the liquid, so that the
diffusion field has a typical “thickness” ofl.
Note that we have not used Eq. (2-26a)
as a boundary condition at infinity, but find
from Eq. (2-32) thatu= –1 implies the so-
called “unit supercooling”, which corre-
sponds to
T(z=•)=T
m–(L
m/c
p) (2-33)
This basically says that the difference in
melting enthalpyL
mbetween liquid and
solid must be compensated by a tempera-
ture difference in order to globally con-
serve the energy of the system during this
stationary process. In other words, if Eq.
(2-33) is not fulfilled by Eq. (2-26a), the
process cannot run with a stationary profile
of the thermal field, Eq. (2-32). On the
other hand, a particular value of the growth
u
z
l
z= exp ;−
⎛
⎝
⎞
⎠
−
2
10Û
12
2
D
u
t
u
l
u
z
T
∂
∂
∇+
∂
∂
=
rateV(orl) is not specified, but seems to
be arbitrary. This degeneracy is practically
eliminated by other effects such as inter-
face kinetics (Collins and Levine, 1985) or
the density difference between solid and
liquid (Caroli et al., 1984), so that in prac-
tice a well-defined velocity will be selected.
b)Growth of a sphere
The influence of the surface tension
g
when it is incorporated through Eq. (2-21)
is most easily understood by looking at a
spherical crystal. This will not lead to sta-
tionary growth. In order to make the analy-
sis simple, we will invoke the so-called
“quasistationary approximation”, by set-
ting the left-hand side in Eq. (2-25) equal
to zero. The physical meaning is that the
diffusion field adjusts itself quickly to a
change in the boundary structure which is,
however, still evolving in time because the
continuity equation is velocity-dependent.
Of course, this approximation reproduces
the stationary solutions precisely (if they
exist), and in addition, it exactly identifies
the instability of a stationary solution as
long as it is not of the Hopf type (i.e., the
critical eigenvalue is not complex). It is
generally assumed that the approximation
is good as long as the diffusion length is
large compared with other lengths of the
evolving pattern.
In a spherical coordinate system, the
equation of motion then becomes
(2-34)
as a simple Laplace equation.
The interface is at radiusR
0(t) and is ad-
vancing with time. The continuity equation
requires
(2-35)
v
RT
Rt
R
t
D
u
r
≡−
∂
∂
d
d
=
=
0
0r ()
1
0
2
2
2
D
u
t rrr
u
T
∂
∂
≈
∂
∂
+
∂
∂
⎧
⎨
⎩
⎫
⎬
⎭
=
92 2 Solidificationwww.iran-mavad.com
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145. 2.2 Basic Concepts in First-Order Phase Transitions 93
Concerning the boundary condition, we
note that for a simple substance the differ-
ence in slopes of the chemical potentials
for solid and liquid at the melting point
(Landau and Lifshitz, 1970) isL
m/T
m, giv-
ing
D
m=–L
m(T–T
m)/T
m (2-36)
With Eqs. (2-21) and (2-24), the boundary
condition in the case of isotropic
gbe-
comes
u(interface) = –d
0K–bV (2-37)
with curvature
(2-38)
capillary length
d
0=gT
mc
p/L
2
m
(2-39)
and interface kinetics
b=k
–1
T
mc
p/L
2
m
(2-40)
The generalization to non-isotropic
gfol-
lows from Eq. (2-21) and comments.
Note that in Eq. (2-37) both curvature
and non-equilibrium occur at the boundary
as opposed to the simple case in Eq. (2-
26b), shown above. At infinity we finally
impose
u
•=u(•,t)=– D (2-41)
as the now arbitrary dimensionless super-
cooling. This equation, together with Eq.
(2-24), is the definition of
D.
The solution to Eq. (2-34) with Eqs. (2-
35), (2-37), and (2-41) in the liquid is then
simply
(2-42)
where we have usedK=2/ras the curva-
ture of the spherical surface in three dimen-
sions.
ur
R
r
d
R
D
R
T
()=−+ −
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟ DD
00
00 2
1 b
K
R
R
=
3-dim
2-dim
2
1
/,
/,
⎧
⎨
⎩
The growth rate therefore is
(2-43)
and it is found that for large radiiR
0(t)the
growth rate is proportional to the super-
cooling
D:
(2-44)
It can also be seen that the kinetic coeffi-
cient
bbecomes less important in the limit
of large radii and correspondingly small
growth rates. This in many cases justifies
our previous equilibrium approximation,
Eq. (2-22).
We have not discussed here the solidifi-
cation of a binary mixture. As the details
of the phase diagram will become impor-
tant later, we will postpone this topic to the
section on directional solidification. We
would like to mention, however, that the
representation of the diffusion field in di-
mensionless form, found in Eq. (2-24), has
the virtue that several results can be carried
over directly from the thermal case to com-
positional diffusion, although there are
some subtleties in the boundary conditions
which make the latter more difficult to an-
alyze.
2.2.4 Mullins–Sekerka Instability
We now combine the considerations of
Secs. 2.2.3a) and b) to study the question
of whether an originally flat interface will
remain flat during the growth process. The
results indicate that a flat interface moving
into a supercooled melt will become rip-
pled (Mullins and Sekerka, 1963, 1964).
The basic equations are almost exactly
the same as in the previous section a) Eqs.
(2-30), (2-27), (2-26a), but instead of the
boundary condition Eq. (2-26b), we now
V
D
R
D
R
R
TT≈−+…
⎛
⎝
⎜
⎞
⎠
⎟
00
1Db
v
R
TT
R
t
D
R
d
R
D
R
≡−
⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟
d
d
=
0
0
0
00 2
1
D
bwww.iran-mavad.com
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146. have to consider deviations from a flat
interface as we move parallel to the inter-
face in thex-direction. This is provided by
Eq. (2-37) as the boundary condition,
where we set
b= 0 for simplicity.
Denoting the deviation of the interface
fromz=0 as
z(x,t), we designate the cur-
vature for small amplitudes simply as
K=–∂
2
z/∂x
2
, where, in agreement with
Eq. (2-38), the sign ofKallows for the cur-
vature to be positive for a solid protrusion
into liquid. We follow here the notation of
Langer (1980a).
In a quasistationary approximation, for
the sake of simplicity we set the l.h.s. in
Eq. (2-30) equal to zero, which of course
allows for the stationary solution, Eq. (2-
32). It is not difficult to treat the fully time-
dependent problem here, but the modifica-
tions do not change the results substan-
tially. We now perturb that solution by
making a small sinusoidal perturbation of
the flat interface:
z(x,t)=z
ˆ
kexp (ikx + W
kt) (2-45)
Similarly, a perturbation of the diffusion
field is made in the liquid and in the solid
where the unprimed form is for the liquid,
and the primed values are for the solid.
Inserting Eq. (2-46) into Eq. (2-30) with
∂u/∂t= 0, we obtain
–2q/l+q
2
–k
2
=0
2q¢/l¢+q¢
2
–k
2
= 0 (2-47)
Replacingzin Eq. (2-46) by
z(x,t)fromEq.
(2-45), we can insert this into the boundary
condition Eq. (2-37) for
b= 0 to obtain
(2-48)
ˆ
ˆ
/ˆ
ˆ
′−+−uludk
kkk k==2
0
2zz
uxzt
z
l
uikxqzt
uxzt u ikx qz t
kk
kk
( , , ) exp
ˆexp ( );
(,,) ˆexp ( )
=(2-4)
=
−
⎛
⎝
⎞
⎠
−
+−+
′′ +′+
2
16
W
W
where we have linearized the exp (…) with
respect to
z. Instead of the velocity in Eq.
(2-27), we must now useV
z=2D/l+∂ z/∂t.
With this equation and the same lineariza-
tion as before
(2-49)
is obtained for the conservation law, Eq.
(2-27) (and for small values of
z,practi-
cally Eq. (2-28)). Eliminatinguˆ
khere using
Eq. (2-48) and eliminatingqusing Eq. (2-47)
in the limitkl∫1ork≈q≈q¢, we obtain
This formula describes the basic mecha-
nism of diffusion-controlled pattern forma-
tion in crystal growth. Althoug in general,
diffusion tends to favor homogeneous
structures, in the present case it works in
the opposite direction! This is easy to
understand; the foremost points of a sinu-
soidal deformation of the interface can dis-
sipate the latent heat of freezing by a larger
space angle into the liquid than the points
inside the bays. The latter points will there-
fore be slowed down. The tip points will be
accelerated in comparison with the average
rate of advancement.
The formula therefore consists of a de-
stabilizing part leading to positive
W
kand
a stabilizing part controlled by the capillar-
ityd
0. The stabilization is most efficient
at largekvalues or short wavelengths; at
longer wavelengths, the destabilization due
to diffusion into the supercooled liquid
dominates.
The dividing line is marked by the com-
bination ofd
0l, which is most conveniently
expressed as
(2-51)
assumingD=D¢. This will be called the
“stability length”, which gives a measure
l
SS==22
0pp/kdl
W
kVq l Dq Dq dk
kV D D ld k
=
(2- )
(/)( )
[( /) ]
−− + ′′
≈−+ ′
2
11 50
0
2
1
2 0
2
W
kk k k kDl DquDqu
ˆ
(/)
ˆ
ˆˆzz=−++ ′′′2
2
94 2 Solidificationwww.iran-mavad.com
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147. 2.3 Basic Experimental Techniques 95
of the typical lengths for diffusion-gener-
ated ripples on the interface, since the
fastest-growing mode has a wavelength
l
f=÷
-
3l
S. This is the first explicit example
of the fundamental scaling relation Eq.
(2-1) mentioned in the introduction to this
chapter.
2.3 Basic Experimental Techniques
Before discussing the various growth
models, a short overview of selected ex-
periments that have contributed to our
present understanding will be given.
An exhaustive treatment of the subject is
not intended here; only some of the charac-
teristic techniques will be described. No
specific reference will be given to the
many solidification processes available.
The main emphasis will be on growth of
crystals, because direct evidence on nucle-
ation mechanisms is generally not avail-
able. There is an exception; the original
technique developed by Schumacher et al.
(1998). These authors analysed quenched-
in nuclei (inoculants) in glassy Al alloys in
the transmission electron microscope. In
this state, nucleation kinetics are very slow
andin-situobservation of the phases and
and their structure, composition and crystal
orientation can be measured. The reader
interested in nucleation experiments is
referred to reviews by Perepezko et al.
(1987, 1996).
There are two essentially different situa-
tions of solidification, or, more generally,
of phase transformation (Fig. 2-1):
i) free (undercooled or equiaxed) growth,
ii) constrained (columnar on directional)
growth.
In the first case (Fig. 2-1a), the melt
undercools, before transforming into a
crystal, until nucleation sets in. The cru-
cible that contains the melt must be less ef-
fectively catalytic to crystallization than
the heterogeneous particles in the melt.
The crystals then grow with an interface
temperature above the temperature of the
surrounding undercooled liquid. Heat is
carried into the liquid. The temperature
gradient at the solid–liquid interface in the
liquid is therefore negative and approxi-
mately zero in the solid owing to the in-
itially small crystal size relative to the ther-
mal boundary layer.
In the second case of constrained growth
(Fig. 2-1b), the temperature gradient is
positive in both liquid and solid, and the
first solid is either formed in contact with
the chill mold or is already present, as in
surface treatment by lasers, for example,
where growth occurs epitaxially after re-
Figure 2-1.Growth in (a) under-
cooled melt (free growth) and (b)
superheated melt (directional
growth). The arrows on the out-
side of the mold represent the heat
flux, and the arrows at the
solid/liquid interface, the growth
direction.www.iran-mavad.com
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148. melting. In case i), nucleation is essential
in controlling the microstructure (the grain
size), while in case ii), nucleation is of mi-
nor importance.
Typical technical processes with respect
to these two classes of solidification are as
follows:
i) casting into low conductivity molds
(ceramics), producing small temperature
gradients, or adding inoculating agents or
stirring the melt, thereby producing many
heterogeneous or homogeneous nuclei;
ii) processes with high heat flux imposed
through strong cooling of the solid such as
continuous casting, welding, laser treat-
ment, or through heating of the melt and
cooling of the solid, such as in single crys-
tal growth orBridgman typedirectional so-
lidification experiments (see also Flem-
mings, 1991).
In free growth, the undercoolingDTof
the melt is given and controls the growth
rateVand the scale (spacing
lor tip radius
R) of the forming microstructure. In direc-
tional growth, the rate of advance of the
isotherms is imposed by heat flux leading
to an imposed growth rate (constrained
growth). This, in turn, controls the inter-
face temperature (undercooling) and the
microstructural scales. These three vari-
ables (temperature, growth rate, and micro-
structural size) have to be measured experi-
mentally for a quantitative comparison of
theory and experiment. A number of mate-
rial parameters of the alloy system have to
be known, such as the solid–liquid interfa-
cial energywith its anisotropy, the diffusion
coefficient, the stable and metastable phase
diagram, etc.
The experimental techniques are divided
into two classes; those which produce rela-
tively small growth rates and those which
aim for high solidification rates. The corre-
sponding experimental setup is quite dif-
ferent, and some of its important elements
will be described below.
2.3.1 Free Growth
Slow growth rates
In free growth, the undercoolingDTor
the temperature of the meltT
0=(T
m–DT)
is imposed on the system, and the growth
rateVand the microstructural scale are the
dependent variables. For low undercool-
ings, and therefore slow growth rates, all
three variables can be measured precisely
if the substance is transparent. The crystal-
lization of non-transparent metals, how-
ever, is the major issue in solidification
studies relevant to technical applications.
Instead of investigating the crystalliza-
tion of metallic systems directly, suitable
model substances have to be found. These
are generally organic “plastic crystals”
which, like metals, have simple crystal
structures and low melting entropies (Jack-
son, 1958). One of the substances which
behaves very similarly to metals and also
has well characterized properties is succi-
nonitrile (SCN) (Huang and Glicksman,
1981). Numerous results have been ob-
tained with this material by Glicksman and
co-workers. Their careful experimental ap-
proach not only produced the most precise
measurements known at that time but has
also stimulated new ideas about possible
mechanisms of structure formation through
the discrepancy found between the obser-
vations and the predictions of previous the-
ories.
Fig. 2-2 shows the experimental setup
developed by Glicksman et al. (1976) to
study free dendritic solidification. After
zone melting, the material is introduced
into the growth chamber (C), which is then
slowly undercooled with the heaters (A)
and (B) in order to avoid premature crystal-
96 2 Solidificationwww.iran-mavad.com
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149. 2.3 Basic Experimental Techniques 97
lization. By careful adjustment of heater
(A), growth starts there, and a crystal
grows through the orifice in chamber (C).
From then on, the dendrite can grow freely
into the undercooled liquid and its shape,
size and growth rate, depending on super-
cooling, are measured with the aid of pho-
tographs. In order to choose a proper pro-
jection plane of observation, the entire
equipment can be translated, rotated and
tilted at (D). A series of similar experi-
ments have been performed at low tem-
peratures with rare gases (Bilgram et al.,
1989).
In this kind of experiment, it is important
to avoid thermal or solutal convection, as
this transport mechanism will change the
results and make them difficult to compare
with diffusional theory. Glicksman et al.
(1988) have shown the effect of thermally
driven convection on the growth morphol-
ogy of pure SCN dendrites. Fig. 2-3 clearly
shows a strong deviation of the growth rate
at undercoolings lower than 1 K, where
convection is believed to accelerate den-
drite growth. HereV
0is the dendrite tip
rate of pure diffusion-controlled growth.
Recent experiments on free dendritic
growth of succinonitrile in space under mi-
crogravity conditions in the undercooling
range of 0.064 to 1.844 K confirmed this
effect clearly (Koss et al., 1999).
Another type of slow dendrite growth in
undercooled media has been analyzed by
Trivedi and Laorchan (1988). These au-
thors made interestingin situobservations
during the crystallization of glasses. Even
if the driving forces in these systems are
very large, growth is heavily restricted due
to slow diffusion and attachment kinetics.
Figure 2-2.Equipment for free growth of organic
dendrites (Glicksman et al., 1976). A and B, control
heaters; C, growth chamber; D, tilting and rotating
device; F, tank cover; G and H, zone-refining tubes.
Figure 2-3.Ratio of measured
growth rate,V, to predicted
rate from diffusion theory,V
D,
as a function of undercooling (Glicksman et al., 1988).www.iran-mavad.com
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150. Fast growth rates
In order to explore solidification behav-
ior under very large driving forces, Herlach
et al. (1993) and Herlach (1994, 1997)
systematically measured dendrite growth
rates in a large number of highly under-
cooled alloys. Nucleation is avoided by
levitation melting in an ultrahigh vacuum
environment. Undercooling of more than
300 K leading to growth rates of up to
70 m/s has been reached. Such large under-
coolings have been obtained by several au-
thors before, but growth rates have not
been measured. Interesting results in Ni–
Sn alloys have also been obtained by high-
speed cinematography of highly under-
cooled samples. These results were pro-
duced by Wu et al. (1987) by encapsulating
the melt in glass. They tried to measure
tip radii from the photograph. In this case,
however, only the thermal images of the
dendrites could be seen, their tips being
controlled by solute diffusion. The radii
from the thermal images are therefore
believed to be much larger than the real
radii.
2.3.2 Directional growth
Slow growth rates
Understanding of solidification improved
in the 1950s when the need for better semi-
conductor materials stimulated research
using directional growth techniques such
as zone melting and Bridgman growth.
Later, directional casting became an impor-
tant topic of research for the production
of single crystal turbine blades. Fig. 2-4
shows the essentials of two techniques:
Bridgman type andchill platedirectional
solidification. The first of these processes
(Fig. 2-4a) has the advantage of being able
to produce a constant growth rate and a
constant temperature gradient over a con-
siderable length and to allow for a certain
uncoupling of these two most important
variables. It was this latter advantage
which, early on in solidification research,
provided much insight into the mecha-
nisms of growth. With one growth rate, but
varying temperature gradient (or vice
versa), plane-front, cellular or dendritic
morphologies could be grown, and their
evolution studied. Important concepts
(constitutional undercooling, cell growth,
98 2 Solidification
Figure 2-4.Basic methods
of directional solidification;
(a) Bridgman type furnace
and (b) directional casting.(a) (b)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

151. 2.3 Basic Experimental Techniques 99
etc.) have been developed with the aid of
observations made with the Bridgman type
of equipment. The experiments by Hunt et
al. (e.g., Burden and Hunt, 1974) deserve
special attention, as the quality of the
measurements was of a very high standard.
Furthermore, most of the work ondirec-
tional eutecticsand their growth mecha-
nisms was performed with the aid of this
technique. In eutectic solidification, the
microstructure after transformation has the
same size (interphase spacing
l) as that
found at the growing interface and thus al-
lows for direct conclusions to be drawn
concerning the growth process. This im-
portant variable can therefore be easily de-
termined in non-transparent metals. This,
however, is not possible with the corre-
sponding quantity of the dendrite, the tip
radiusR, as discussed earlier.
One disadvantage of the Bridgman ex-
periment is the need for a small diameter
due to heat flux constraints. This is avoided
in the process shown in Fig. 2-4b (direc-
tional casting), but in this case a separation
of the variablesVandGis not possible.
Therefore, this method is less interesting
for scientific purposes. However it is used
extensively for directional casting of single
crystal turbine blades (Versnyder and
Shank, 1970).
Forin-situobservation of microstructure
information, we can also use plastic crys-
tals in an experimental arrangement resem-
bling a two-dimensional Bridgman appara-
tus, as shown in Fig. 2-5. Two glass slides
containing the organic analogue are moved
over a heating and cooling device produc-
ing a constant temperature gradientG
T.
This can be controlled to certain limits by
the temperature difference and distance
between heater and cooler (Somboonsuk
and Trivedi, 1985; Trivedi and Somboon-
suk, 1985; Akamatsu et al., 1995; Ginibre
et al., 1997; Akamatsu and Faivre, 1998).
A thin thermocouple incorporated into
the alloy allows for measurement of inter-
face temperature (Fig. 2-6) and tempera-
ture gradient in liquid and solid (Esaka and
Kurz, 1985; Somboonsuk and Trivedi,
1985; Trivedi and Somboonsuk, 1985; Tri-
vedi and Kurz, 1986).
Fast growth rates
There have been attempts to increase the
growth rate in Bridgman type experiments.
The first to reach rates of the order of sev-
eral mm/s were Livingston et al. (1970).
The best way to reach much higher rates is
through laser resolidification using a stable
Figure 2-5.Schematic diagram of growth cell (a, b)
for observation of directional solidification of trans-
parent substances under a microscope (Esaka and
Kurz, 1985). 1, solid; 2, liquid; 3, thermocouple;
4, cell; 5, low melting glass; 6, araldite seal; 7 and 8,
heaters; 9, cooler; 10, drive mechanism; 11, micro-
scope. In (c) the temperature distribution in the
growth cell is shown.www.iran-mavad.com
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152. high-powered laser. In this case, a small
melt pool with very steep temperature gra-
dients is created (for a remelted layer of
100 mm thickness, the temperature gradi-
entG
Tis of the order of 5000 K/mm). The
microstructure is then constrained to fol-
low closely the heat flow, which is perpen-
dicular to the isotherms (Fig. 2-7). There-
fore, knowing the angle
qbetween the di-
rection of growth and the direction and rate
of movement of the laser beam alloys for
the growth rate to be obtained locally, even
in the electron microscope (Zimmermann
et al., 1989). Note that the growth rate in-
creases from zero at the interface with the
substrate up to a maximum at the surface.
The only unknown which would be ex-
tremely difficult to measure is the interface
temperature.
Consequently, in the case of rapid direc-
tional growth we can determine the post-
solidification microstructure and its growth
rate as well as the bath temperature and the
growth rate in rapid undercooled growth.
In both cases, the measurement of the inter-
face temperature is not possible for the
time being and must be evaluated through
theory alone.
2.4 Free Dendritic Growth
The most popular example of dendriti-
cally (= tree-like) growing crystals is given
by snowflakes. The six primary arms on
each snowflake look rather similar but there
is an enormous variety of forms between
different snowflakes (Nakaya, 1954). From
our present knowledge, the similarity of
arms on the same snowflake is an indica-
tion of similar growth conditions over the
arms of a snowflake, the variation of struc-
ture along each arm being an indication of
a time variation of external growth condi-
tions.
In order to formulate a theory, some spe-
cific assumptions about these environmen-
tal conditions must be made. Unfortu-
nately, the growth of a snowflake is an
extremely complicated process, involving
strongly anisotropic surface tension and
kinetics and the transport of heat, water va-
por, and even impurities. Therefore, we
will initially drastically simplify the model
assumptions. In the same spirit, a number
of precise experiments have been per-
formed to identify quantitatively the most
important ingredients for this mechanism.
In its simplest form, dendritic growth re-
quires only a supercooled one-component
liquid with a solid nucleus inside, so that
the heat released at the solid–liquid inter-
face during growth is transported away into
the liquid. This is precisely the condition
given in Sec. 2.2.3 concerning a sphere
growing into the supercooled, and there-
100 2 Solidification
Figure 2-6.Array of dendrites approaching a fine
thermocouple for measurement of the tip temperature
and temperature gradient. Diameter of the bead ap-
proximately 50 µm (Esaka and Kurz, 1985).www.iran-mavad.com
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153. 2.4 Free Dendritic Growth 101
fore metastable, liquid. A relatively straight-
forward stability analysis (Mullins and Se-
kerka, 1963, 1964; Langer, 1980a) shows
that the solid tends to deviate from spheri-
cal form as soon as its radiusRhas become
larger than seven times the critical radius
R
c=2d
0/D. A time-dependent analysis has
also been carried out (Yokoyama and Ku-
roda, 1988). During the further evolution,
these deformations evolve into essentially
independent arms, the primary dendrites.
The growth of such dendrites is a very
widespread phenomenon, as will become
clearer in Sec. 2.5 on directional solidifica-
tion. There are also close relations to pro-
cesses called diffusion-limited aggregation
(DLA) (for a review see Meakin, 1988; for
relation to crystal growth, see Uwaha and
Saito, 1990; Xiao et al., 1988). For these
processes, some specific assumptions are
made about the incorporation of atoms into
the advancing interface which are not eas-
ily carried over into the notion of surface
tension, etc. Another line of closely related
problems concerns the viscous flow of two
immiscible liquids (Saffmann and Taylor,
1958; Brener et al., 1988; Dombre and Ha-
kim, 1987; Kessler and Levine, 1986c). We
will briefly refer to this in Sec. 2.5.
2.4.1 The Needle Crystal Solution
In this subsection, we will look at an iso-
lated, needle-shaped crystal growing under
stationary conditions into a supercooled
melt. A stationary condition, of course, only
Figure 2-7.Schematic diagrams of a la-
ser trace (a, b) and of the local interface
velocity (c).V
bis the laser scanning ve-
locity andV
sis the velocity of the solid/
liquid interface which increases from zero
at the bottom of the trace to a maximum
at the surface.WandDare the width and
depth of the trace, respectively, and
qis
the angle between the growth direction
and the scanning direction.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

154. holds in a frame of reference moving at ve-
locityVin the positivez-direction. For a one-
component crystal, the latent heat of freez-
ing must be diffused away into the liquid.
The process is therefore governed by the
heat diffusion Eq. (2-25)
(2-52)
where we have used a dimensionless form,
Eq. (2-30), for a stationary pattern together
with the definition of the diffusion length,
Eq. (2-31). If the diffusion constants in the
crystal and the liquid are different, we
would have to use two diffusion lengths.
The continuity Eq. (2-27) at the interface is
for normal velocity
V
^=–D
Tnˆ·—u
L+D
Tnˆ·—u
S (2-53)
where the subscripts L and S denote gra-
dients taken on the liquid side and on the
solid side of the interface. This is known as
the “two-sided” model (Langer and Turski,
1977) and, because of identical diffusion
constants, as the “symmetrical” model. The
generalization to different diffusion con-
stants is simple.
The boundary condition at the interface is
u
I=–d
0K–bV (2-54)
and at infinity
u
•=–D (2-55)
as introduced before in Eqs. (2-37) and (2-
41), respectively. Note that a constant like
Dmay be added on the right-hand side of
both equations without changing the results
apart from this additive constant in theu-
field, as frequently appears in the litera-
ture. The definition of
Dfollows easily
from Eqs. (2-24) and (2-41).
In general, the capillary lengthd
0and
the kinetic coefficient
bare anisotropic be-
cause of the anisotropy of the crystalline
0
2
2
=∇+
∂
∂
u
l
u
z
lattice, but they are not directly related (Burkhardt et al., 1977). For the moment, we will assume
b= 0, i.e., the interface ki-
netics should be infinitely fast. Even then, the curvatureKin Eq. (2-54) is a compli-
cated function of the interface profile. The simplest approximation for the moment, therefore, is to ignore both terms on the r.h.s. of the boundary condition, which cor- responds to setting the surface tension to zero. The boundary condition is then sim- ply a constant (= 0 in our notation).
A second-order partial differential equa-
tion such as the diffusion equation can be decomposed in the typical orthogonal coor- dinate systems, and we therefore obtain a closed form solution for the problem with a boundary of parabolic shape: a rotational paraboloid in three dimensions and a simple parabola in two dimensions. This is the famous “Ivantsov” solution (Ivantsov, 1947).
The straightforward way to look at this
problem is as a coordinate transformation from the cartesian {x,z} frame, wherezis
the growth direction, to the parabolic coor- dinates {
x,h}:
x=(r–z)/ r;h=(r+z)/ r
whereris the radial distance÷
---
x
2
+
---
z
2
from
the origin, and
ris a constant. After trans-
forming the differential operators in Eqs.
(2-52) and (2-53) to {
x,h} (Langer and
Müller-Krumbhaar, 1977, 1978), it can be
seen immediately that
h(x)=h
S=1 for the
interface is a solution to the problem con-
firming the parabolic shape of the inter-
face, with
rbeing the radius of curvature at
the tip.
ThisIvantsov radius,
r, is now related to
the supercooling
Dand the diffusion length
lby the relation
(2-56a)
(2-56b)
D=
e 3-dim
e erfc 2-dim
PEP
PP
P
P
1
()
()p
⎧
⎨
⎩
102 2 Solidificationwww.iran-mavad.com
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155. 2.4 Free Dendritic Growth 103
which for smallD≤1 gives
and for
DÆ1 asymptotically
where the dimensionlessPéclet number P
is defined as the relation
P=
r/l=rV/2D (2-57)
between tip radius and growth rate.E
1(P)
is the exponential integral
(2-58)
and erfc is the complement of the error
function.
Eq. (2-56) may be interpreted as an ex-
pression of supercooling in terms of the
Péclet number. This explanation is impor-
tant because the following considerations
of surface tension give only very small cor-
rections to the shape of the needle crystal.
Therefore, Eq. (2-56) will be also valid
with non-zero surface tension at typical ex-
perimental undercoolings of
DÓ10
–1
.An
important consequence is that the scaling
results derived below then become inde-
pendent of the dimensionality (2 or 3), if
the supercooling is expressed through the
Péclet number.
The basic result of Eqs. (2-57) and (2-
48) is that the growth rate of this parabolic
needle is inversely proportional to the tip
radius, but no specific velocity is selected.
For experimental comparison, we make a
fit of the tip shape to a parabola. The tip
radius of that parabola is then compared
with the Ivantsov radius
r. The actual tip
radius will be different from the Ivantsov
radius of the fit parabola, because of sur-
EP
t
t
P
t
1
()=
e
d
∞−
∫
D≈
−
−
⎧
⎨
⎩
11
112
/
/
P
P
3-dim
2-dim
D≈
−− …⎧
⎨
⎩
PP
P
(ln . )0 5772 3-dim
2-dimp
face tension effects. Before we consider
surface tension explicitly we now give an
integral formulation for the problem using
Green’s functions, which has proved to be
more convenient for analytical and numer-
ical calculations (Nash and Glicksman,
1974).
The value of a temperature fieldu(x,t)
in space and time is obtained by multiply-
ing the Green’s function of the diffusion
equation with the distribution for sources
and sinks for heat and integrating the prod-
uct over all space and time. In our case, this
explicitly gives
(2-59)
(Langer, 1987b), with the Green’s function
(2-60)
for diffusion in an infinite three-dimen-
sional medium (symmetrical case), while
z
is thezcoordinate of the interface. The
source term {…} in this equation is obvi-
ously the interface in the frame of refer-
ence moving at velocityVin thez-direc-
tion. This equation already contains the
conservation law or the continuity equation
at the interface. Furthermore, it is valid
everywhere in space and, in particular, at
the interfacez=
z(x,t), where the l.h.s. of
Eq. (2-59) is then set equal to Eq. (2-54).
Hereuis assumed to vanish at infinity, so
Dmust be added on the l.h.s.
In the two-dimensional case and for
stationary conditions, this can be rewritten
as
(2-61)
DG−
d
KPPxx
0
2
r
zz
{} { , , ( )}=
Gzt
Dt
z
Dt
(,,)
()
exp
/
x
x
=
1
4 4
32
22
p
||
−
+⎛
⎝
⎜
⎞
⎠
⎟
uzt t G z t
Vt t t t V
t
t
(,,) [ , ( , )
(), ]
xxx xx=d d
−∞
∫∫′′ −′−′′
+− ′−′+
∂
∂′
⎧
⎨
⎩
⎫
⎬
⎭ z
zwww.iran-mavad.com
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156. (Langer, 1987b), with
In addition, it was shown (Pelce and Po-
meau, 1986) that with the parabolic Ivant-
sov solution
(2-63)
we can obtain
which is independent ofx, with
Dcoming
from Eq. (2-56b). Note that from Eqs.
(2-59) to (2-61) we have replacedxbyx
r
etc., which mirrors the scaling form of
Eq. (2-57).
We are now ready to consider a non-zero
capillary lengthd
0Eq. (2-39), which we
generalize to be anisotropic:
d
0Æd=d
0(1 –ecos 4J) (2-64)
where
Jis the angle between the interface-
normal and thez-axis (2-dim, 4-fold an-
isotropy), and
e> 0 is the relative strength
of that anisotropy. This form arises from
the assumption
g=g
0(1 +dcos 4J) (2-65)
for anisotropy surface tension, which gives
through Eq. (2-22)
e=15d. Note that it is
the stiffness
g≤which dominates the be-
havior, not
gdirectly. Along the Ivantsov
parabola, the angle
Jis related tox:
(2-66)
and the deviation from
z
IV(x) can be ex-
pressed as
(2-67)
−−
d
P
KPxPx
r
zzz
() {, , } {, , }=
IVGG
22
d d Ax Ax
x
x
==
0
2
22 1
8
1
(); ()
()
−+
+e
e
PPx xGD
2{, , (z
IV)} =
zz()xx→−
IV=
1
2
2
G
2
0
22
1
2
2
{, , ()}
exp {( ) [ ( ) ( ) ] }
Px x
y
y
x
P
y
xx x x y
z
zz=
d
d (2-62)
p
∞
−∞
+∞
∫∫ ′
×− − ′+− ′+
⎛
⎝
⎜
⎞
⎠
⎟
For convenience we combine some param-
eters into a dimensionless quantity
s:
(2-68)
so that the l.h.s. of Eq. (2-67) becomes
(2-69)
the curvature as usual being
(2-70)
It should be clear at this point that the pa-
rameter
sin Eq. (2-68) plays an important
role, because it multiplies the highest de-
rivative in Eqs. (2-61) and (2-67). More
specifically, we can expect that the result-
ing structure
z(x) of the interface depends
on the material properties and experimental
conditions only through this parameter
s
(within the model assumptions), which be-
comes a function ofPand
e
s
=s(P,e) (2-71)
The importance of the paramter
swas
recognized in an earlier stability analysis
(Langer and Müller-Krumbhaar, 1978,
1980) of the isotropic case. It turned out
later, however, that the anisotropy is essen-
tial in determining the precise value of
s.
This is crucial as
eÆ0 impliessÆ0, i.e.,
no stationary needle solutions exist without
anisotropy.
We will now briefly describe the analy-
sis of Eq. (2-67). The details of this singu-
lar perturbation theory are somewhat in-
volved, and we therefore have to omit them
here. The basic method was formulated by
Kruskal and Segur (1985), and the first
scaling relations for dendritic growth were
obtained for the boundary-layer model
(Ben Jacob et al., 1983, 1984). A good in-
troduction to the mathematical aspects can
K
x
x
{}
/
[(/)]
/
z
z
z=−
∂∂
+∂ ∂
22
232
1
−−
d
P
KAK
r
s
=
s
r==
2
2
0
2
0
2Dd
V
d
DP
V
104 2 Solidificationwww.iran-mavad.com
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157. 2.4 Free Dendritic Growth 105
be found in the lectures by Langer (1987b),
on which the following presentation is
based. The most mathematically sound
(nonlinear) solution seems to have been
given by Ben Amar and Pomeau (1986).
For convenience, we have sketched a
slightly earlier linearized version here,
while the nonlinear treatment leads to basi-
cally the same result.
Looking for a solution to Eq. (2-67) in
linear approximation, we start by expand-
ing to first order in
z
1(x)=z(x)–z
IV(x) (2-72)
In the limit of the small Péclet number, and
with the substitution
z
1(x)=(1+x
2
)
3/4
Z(x) (2-73)
gives
(
Bˆ+Aˆ)·Z(x)= s/(1 +x
2
)
3/4
(2-74)
where
Bˆis a self-adjoint differential opera-
tor
(2-75)
and
with∫denoting the principal value.
The integral kernel in Eq. (2-76) is anti-
symmetric apart from a prefactorA(x)
–1
.
An analytic solution to Eq. (2-74) has not
yet been found. A necessary condition to
be fulfilled by the present inhomogeneity is
that it should be orthogonal to the null-
eigenvectorsZ
˜
(x) of the adjoint homoge-
neous problem:
(
Bˆ+Aˆ
+
)·Z(x) = 0 (2-77)
ˆ
()
()
()
()( )
()[()]
()
/
/
A⋅
+
× ′
+′+′
−′++ ′
′
−∞
+∞
∫
Zx
x
Ax
x
xx x
xx xx
Zx
= (2-76)
d
1
2
1
1
234
234
1
4
2
p
∫
ˆ
()
()
()
/
B=
d
dss
2
2
212
1
0
x
x
Ax
+
+
+
such that
(2-78)
In fact, this is already a sufficient condition
for the solvability of the inhomogeneous
equation, but it is not very simple.
A solution forZ
˜
(x) can be found by a
WKB technique, for which we refer to the
literature (Kessler et al., 1987, 1988; Lan-
ger, 1987b; Caroli et al., 1986a, b). The
result for the solvability condition, Eqs.
(2-71) and (2-78), is that the parameter
s
should depend on anisotropyeas
s≈s
0e
7/4
(2-79)
in the limitPÆ0,
eÆ0, with some con-
stant prefactor
s
0of order unity.
Eq. (2-79) is the solution for the needle-
shaphed crystal with capillary anisotropy
e> 0, together with Eq. (2-68). Note again
that
eis the anisotropy of capillary length,
which differs by a factor from surface-ten-
sion anisotropy Eq. (2-65). Formally, there
is not just one solution but infinitely many,
corresponding to slow, fat needles which
are dynamically unstable. Only the fastest
of these needle solutions appears to be
stable against tip-splitting fluctuations and
may thus represent an ‘observable’ needle
crystal, as expressed by Eq. (2-79). For
practical comparison, experimental data is
best compared with numerical solutions of
Eq. (2-61), because the applicability of Eq.
(2-79) seems to be restricted to very small
values of
e(Meiron, 1986; Ben Amar and
Moussallam, 1987; Misbah, 1987). This will
be discussed further in the next section.
Needle-crystal solution
in three-dimensional dendritic growth
The theory of dendritic growth becomes
extremely difficult, however, for three-di-
L(,)
˜
()
()
/
se≡ ′
+
−∞
+∞
∫d=x
Zx
x1
0
234www.iran-mavad.com
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158. mensional (3D) anisotropic crystals. A
simple extrapolation of the 2D case, where
the surface energy is averaged in the azi-
muthal direction (axisymmetric approach,
Ben Amar, 1998; Barbieri and Langer,
1989), is very important in order to have
some qualitative predictions. But any phys-
ical anisotropy will give rise to a non-
axisymmetric shape of the crystal. A nu-
merical approach to the non-axisymmetric
problem was presented by Kessler and Le-
vine (1988) who pointed out the following
aspect of the problem. In either the 2D or
the axisymmetric case, selection of the
growth velocity follows from the solvabil-
ity condition of smoothness of the dendrite
tip. In the 3D non-axisymmetric case a
solvability condition must be satisfied for
each of the azimuthal harmonics. Kessler
and Levine made several approximations
and performed only a two-mode calcula-
tion, but the crucial point of their analysis
is that they found enough degrees of free-
dom to satisfy all solvability conditions.
More recently, an analytic theory of
three-dimensional dendritic growth has
been developed by Ben Amar and Brener
(1993). In the framework of asymptotics
beyond all orders, they derived the inner
equation in the complex plane for the non-
axisymmetric shape correction to the Ivant-
sov paraboloid. The solvability condition
for this equation provides selection of both
the stability parameter
sµe
7/4
and the
interface shape. The selected shape can be
written as
(2-80)
where all lengths are reduced by the tip ra-
dius of curvature
r. Solvability theory (Ben
Amar and Brener, 1993) predicts that the
numbersA
mare independent of the aniso-
tropy strength
a, in the limit of smalla.
For example, the first non-trivial term for
zr
r
Ar m
m
m( , ) cosff=−+ ∑
2
2
cubic symmetry corresponds tom=4 and
A
4= 1/96 is only numerically small. There-
fore, the shape correction, Eq. (2-80), in units of the tip radius of curvature, depends mostly on the crystalline symmetry and is almost independent of the material and growth parameters.
Stability theory for the 3D dendritic
growth against tip-splitting modes has been developed by Brener and Mel’nikov (1995).
An important aspect of Eq. (2-80) is that
the shift vectorr
m
cosmfgrows at a faster
rate than the underlying Ivantsov solution. This means that only the tip region, where the anisotropy correction is still small, can be described by the usual approximation (Kessler and Levine, 1988; Ben Amar and Brener, 1993), a linearization around the Ivantsov paraboloid. This is the crucial dif- ference between the 3D non-axisymmetric case and the 2D case. In the latter, small anisotropy implies that the shape of the se- lected needle crystal is close to the Ivant- sov parabola everywhere; in the former, strong deviations from the Ivantsov para- boloid appear for any anisotropy.
Several important questions arise. How
is the tail of the dendrite to be described? Is it possible to match the non-axisymmetric shape (Eq. (2-80)) in the tip region to the asymptotic shape in the tail region? What is the final needle-crystal solution? The an- swers to these questions have been given by Brener (1993).
The basic idea is that the non-axisym-
metric shape correction, generated in the tip region, should be used as an ‘initial’ condition for a time-dependenttwo-dimen-
sionalproblem describing the motion of
thecross-sectionof the interface in the tail
region. In this reduced description, the role of time is played by the coordinatezfor
steady-state growth in thezdirection. The
deviation from the isotropic Ivantsov solu- tion remains small only during the initial
106 2 Solidificationwww.iran-mavad.com
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159. 2.4 Free Dendritic Growth 107
period of the evolution. This initial devia-
tion can be handled by a linear theory start-
ing from a mode expansion which takes the
same form as Eq. (2-80) with, in principle,
arbitrary coefficients. These amplitudes
then have to be chosen according to the
predictions of selection theory (Ben Amar
and Brener, 1993) in order to provide the
matching to the tip region. As ‘time’ goes
on, the deviation increases owing to the
Mullins–Sekerka instability and a nonlin-
ear theory must take over. We can guess
what the long-time behavior of the system
will be. Indeed, this two-dimensional prob-
lem is precisely the same as that which
leads to two-dimensional dendritic struc-
tures. Four well-developed arms (for cubic
symmetry) grow with a constant speed in
the directions favored by the surface en-
ergy anisotropy. Each arm has a parabolic
shape, its growth velocity,v
2=2DP
2
2(D)
¥
s
2
*(e)/d
0, and the radius of curvature of
its tip,
r
2=d
0/(P
2s
2
*), are given by 2D se-
lection theory, where the anisotropic sur-
face energy again plays a crucial role. The
Péclet numberP
2=r
2v
2/2Dis related to
the undercooling
Dby the 2D Ivantsov for-
mula, which for small
Dgives
P
2=(D)=D
2
/p (2-81)
The selected stability parameter
s
2=
d
0/(P
2r
2) depends on the strength of the
anisotropy
e, ands
2
*(e)µe
7/4
for smalle
as was explained above. Replacingtby
|z|/vand reducing all lengths by
r,wecan
present the shape of one-quarter of the
interface (except very close to the dendritic
backbone) in the form
(2-82)
This asymptotic describes the strongly an-
isotropy interface shape far behind the tip
and does not match, for small
D, the shape
described by Eq. (2-80). In this case an im-
xyz z
y
(, )=||
v
v
2
2
2
2
−
r
r
portant intermediate asymptotic exists if the size of the 2D pattern is still much smaller than a diffusion length (Dt)
1/2
.
This 2D Laplacian problem (with a fixed flux from the outside) was solved, both nu- merically and analytically, by Almgren et al. (1993) who were interested in aniso- tropic Hele–Shaw flow. They found that after some transition time the system shows an asymptotic behavior which is in- dependent of the initial conditions and in- volves the formation of four well-devel- oped arms. The length of these arms in- creases in time ast
3/5
and their width in-
creases ast
2/5
. The basic idea that explains
these scaling relations is that the stability parameter
s
2=2Dd
0/(r
2
2v
2) is supposed to
be equal to
s
2
*(e) even though bothv
2and
r
2depend on time. Moreover, the growing
self-similar shape of the arms was deter- mined. In terms of the dendritic problem it reads (Brener, 1993)
(2-83a)
where the tip positionx
tipof the arm is
given by
x
tip(z)=(5|z|/3)
3/5
(s
2
*/s*)
1/5
(2-83b)
The ratio
s
2
*(e)/s*(e) is independent ofe
in the limit of smalle. This means that
the shape, Eq. (2-83), in the tail region is
almost independent of the material and
growth parameters, as well as shape, Eq.
(2-80), in the tip region (if all lengths are
reduced by
r).
Recent experiments on 3D dendrites
(Bisang and Bilgram, 1995) and numerical
simulations (Karma and Rappel, 1998) are
in very good agreement with these theoret-
ical predictions.
yxz z
x
x
s
ss
xx
(,) ( /)
*
/
/ /
/
/
*
=
d
tip
tip
53
1
25
2
15 23
1
23 4
||
s
s
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
×
−
∫www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

160. 2.4.2 Side-Branching Dendrites
This section provides a summary of the
present understanding of dendritic growth.
It is centered around numerical simulations
of isolated side-branching dendrites in a
one-component system where heat diffu-
sion is the relevant dynamical process.
Alternatively, it also describes dendritic
growth from a two-component system at
essentially constant temperature. In the lat-
ter case, we should also discuss the phase
diagram; this will be covered later in the
section on directional solidification. For
many typical cases of growth from a dilute
solution however, the information con-
tained in this section should be sufficient.
We start with the definition of the model
resulting from the set of Eqs. (2-52) to (2-
55). The dynamics come from the conser-
vation law, Eq. (2-53), at the interface. As
in the previous section, we use here the
form of dimensionless units introduced in
Sec. 2.2.3 for the case of heat diffusion.
The case of chemical diffusion (tempera-
ture then being assumed constant =T
0) can
be treated by the same equations. The nor-
malization is described in Sec. 2.5.2. For
convenience, we will simply summarize
here the basic formulas for both cases.
In contrast to Eqs. (2-37) and (2-54), we
will now normalize the following equation
to obtainu= 0 at infinity, which results
from adding the constant
Dto the field-var-
iableuin all equations. Then, the diffusion
field becomes
(2-84)
where
mis the chemical potential differ-
ence between solute and solvent andDC
(0≤DC≤1) is the miscibility gap at the op-
erating temperatureT
0.
u
TT Lc
CC
p
=
thermal diffusion
chemical diffusion
m()/()
()/(/)
−
−
⎧
⎨
⎪
⎪
⎩
⎪
⎪
∞
−
∞
1
mm mD∂ ∂
The dimensionless supercooling is given
as
(2-85)
The capillary length is then
The quantity∂
m/∂Cis not easily measured,
but for smallDC≤1 of a dilute solution, it
can be related to the slope of the liquidus
line dT/dCatT
0=T
mby
(2-87)
(Mullins and Sekerka, 1963, 1964; Langer,
1980a). Note that the chemical capillary
length can be several orders of magnitude
larger than the thermal length.
The boundary condition, Eq. (2-54), then
simply changes to
u
I=D–dK– bV (2-88)
where now the anisotropic capillary length
dis used. The kinetic coefficient may also
depend on concentration (Caroli et al.,
1988), which we ignore here. Far away
from the interface in the liquid, the bound-
ary condition becomes
u
•= 0 (2-89)
For the chemical case, we may practically
ignore diffusion in the solid.
The diffusion Eq. (2-52) and the conser-
vation law, Eq. (2-53), remain unchanged,
and the diffusion length is defined as be-
fore asl=2D
T/V, withVbeing the average
velocity of the growing dendrite.
∂
∂
≈m
C
L
TC
T
C
m
md
dD
d
TcL
CC
p
=
thermal (2-86)
chemical
mm
[() ()]
(/)gg
m
JJ+′′
⎧
⎨
⎪
⎪
⎩
⎪
⎪
−
−−
2
21
D∂∂
D=
thermal
chemical
mm
eq()/()
()/(/)
TT Lc
CC
p−
−
⎧
⎨
⎪
⎪
⎩
⎪
⎪
∞
−
∞
1
mm m D∂ ∂
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161. 2.4 Free Dendritic Growth 109
There are, of course, important differ-
ences between two and three dimensions,
as a three-dimensional needle crystal is not
necessarily rotationally symmetric around
its axis. Snow crystals, for example, show
large anisotropies in directions orthogonal
to the growth direction of the primary den-
dritic needle (Yokoyama and Kuroda,
1988). In the immediate neighborhood of
the tip, however, the deviation from this ra-
tional symmetry is often small. Therefore,
we may work with this two-dimensional
model by using an “effective” capillary
length. The scaling relations given below
are expected to be insensitive to this apart
from a constant prefactor of order unity in
the
s(e)-relation (Kessler and Levine,
1986b, d; Langer, 1987a).
The numerical simulations were per-
formed for a two-dimensional crystal–liq-
uid interface. In Fig. 2-8, we show a typical
dendrite with side branches resulting from
the time-dependent calculations (Saito et
al., 1987, 1988) (compare with the experi-
mental result by Glicksman et al. (1976) in
Fig. 2-9). The profile is symmetric around
the axis by definition of the calculation. An
approximately parabolic tip has been
formed from which side branches begin to
grow further down the shaft (only the early
stage of side-branch formation was consid-
ered). They have a typical distance which,
however, is not strictly regular.
As a first result, the scaling relation, Eq.
(2-68), was checked using the Péclet num-
ber from Eq. (2-56b). Experimentally, this
requires the anisotropic capillary length
and the supercooling to be varied indepen-
dently. In Fig. 2-10, the scaled numerical
results are shown as symbols for two dif-
Figure 2-8.Free dendrite in stationary growth com-
puted in quasistationary approximation for the two-
dimensional case. Capillary anisotropy was
e=0.1
(Saito et al., 1988). The parameter-dependence of the
growth rate, tip radius and sidebranch spacing is con-
sistent with analytical scaling results from solvability
theory of the needle crystal.
Figure 2-9.Dendrite tip in pure succinonitrile (SCN)
at small undercoolings and inscribed parabola for measuring the tip radius (Huang and Glicksman, 1981).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

162. ferent supercoolings and compared with
the results (full lines) for the stationary
needle crystal. The upper line corresponds
to the model with diffusion in the liquid
only (Misbah, 1987), as used in the numer-
ical simulation here. The lower line corre-
sponds to the symmetrical model with
equal diffusion in liquid and solid (Ben
Amar and Moussallam, 1987). Ap-
parently, the two results look the same,
apart from a factor of approximately two
in
s. Note that in unscaled form (i.e., mul-
tiplying byP
2
) the data for the two super-
coolings would differ by about two orders
of magnitude!
From an experimental point of view, it is
better to use Fig. 2-10 rather than Eq. (2-
79) for comparison, as the range of validity
of Eq. (2-79) seems to be restricted to
rather small values of
e. For unknown ma-
terial parameters such as diffusion con-
stant, capillary length and anisotropy, we
can still check the scaling relation of the
growth rateVthrough the Péclet number,
Eq. (2-56), depending on supercooling. Eq.
(2-68) should then give a constant, al-
though arbitrary, value of
s. This scaling
result was confirmed experimentally in
the 1970s, before the full theory existed
(Langer and Müller-Krumbhaar, 1978,
1980). At that time, it was assumed (“mar-
ginal stability” hypothesis) that a universal
value of
s≈0.03 was determined by a dy-
namic mechanism independent of aniso-
tropy. The results for the needle crystal, to-
gether with these numerical simulations,
now show that
sdepends on anisotropye
as shown in Fig. 2-10. Experimental tests
on the
e-dependence (Sec. 2.4.3) are still
rather sparse and do not quite fit that pic-
ture, for reasons not well understood.
So far we have only looked at the rela-
tion between growth rate, anisotropy and
supercooling. We will now consider the
size of the dendrite, which is approxi-
mately parabolic, and which can probably
be characterized by the radius of curvature
at its tip.
This is a subtle point, as the tip radius
cannot easily be measured directly. As an
alternative, we can try to fit a parabola to
the observed dendrite in the tip region. The
tip radius of this fitted parabola should be
interpreted as the Ivantsov radius
r, which
turns out to be slightly larger than the true
tip radiusRof the dendrite. The deviation
ofRfor
rdoes not depend on supercooling
Dbut on anisotropye. This is shown in Fig.
2-11, where a comparison is made between
the dynamic numerical simulations (Saito
et al., 1988) and the needle crystal solution
in the limit of small Péclet number (Ben
Amar and Moussallam, 1987). It can be
seen that there is excellent agreement and
that the actual tip radiusRbecomes smaller
than the Ivantsov radius
rat increasinge.
We now can relate the growth rateVand
the tip radiusRor the Ivantsov radius
rin
110 2 Solidification
Figure 2-10.Scaling parameter s(e) for free den-
dritic growth depending on capillary anisotropy
e
and for two-dimensional supercoolingsD. Average
capillary length isd
0, diffusion constantD, and
Péclet numberP. Comparison of numerical results
(circles and squares, Fig. 2-8, one-sided model) with
solvability results: (a) one-sided model (Misbah,
1987), (b) two-sided model (Ben Amar and Moussal-
lam, 1987). The agreement is excellent, the indepen-
dence upon supercooling is seen to work at least up
to
D= 0.5.www.iran-mavad.com
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163. 2.4 Free Dendritic Growth 111
order to check the scaling form Eq. (2-68)
involving the radius rather than the Péclet
number. The Ivantsov parabola and its ra-
dius
rbasically originate from a global
conservation law for the quantity (heat) re-
leased at the interface, while the tip radius
Ris a local geometric quantity. In practical
experiments, by fitting a parabola to the
tip, we can interpolate between these two
numbers, the result depending on how far
down the shaft the fitting parabola is used.
Using the actual radiusRrather than the
Ivantsov radius
r, perfect scaling can be
seen in Fig. 2-12 with respect to supercool-
ing
D, even up to the very large value of
D= 0.5. Since for smaller supercoolings,
DÙ0.1, the difference betweenrandRbe-
comes negligible, as shown in Fig. 2-11,
and we may safely use Eq. (2-68) as
(2-90)
independent of supercooling
D, to interpret
experiments and to make predictions. The
term “constant” here means that the prod-
uctVR
2
depends on material parameters
VR V
Dd
22 0 2
≈
r
se= = constant
()
only. This is precisely the relation, Eq. (2- 1), derived from qualitative considerations in the introduction to this chapter. This re- lation has been confirmed by the analysis of many experiments (Huang and Glicks- man, 1981).
The final point to be discussed here con-
cerns the side branches and their origin, spacing and amplitudes. This issue is theo- retically not completely resolved, because none of the available analytical approxima- tions can correctly handle the long-wave- length limit of side-branch perturbations. Moreover, the subject is somewhat tech- nically involved. Therefore, we will only summarize the main arguments below and refer to the above-mentioned numerical simulations (Saito et al., 1988) for compar- ison with experiments.
An important quantity which charac-
terizes the stability of flat moving interface ripples is the so-called stability length
(2-91)
whered
0is the capillary length, andlthe
diffusion length. Perturbations of wave-
l
S=2
0pld
Figure 2-11.Tip radius of free dendrite over Ivant-
sov radius plotted versus anisotropy as a function of
dimensionless supercooling. The numerical results
(see also Fig. 2-8, 2-10) are consistent with the pre-
dictions from the needle solution (Ben Amar and
Moussallam, 1987).
Figure 2-12.Numerical scaling result confirming
VR
2
= const. for free dendritic growth independent
of supercooling
D, depending on anisotropyeonly
(Saito et al., 1988).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

164. lengthsllonger thanl
Swill grow, while
shorter wavelengths will decay with time.
This quantity characterizes the competition
between the destabilizing diffusion field
throughlagainst the stabilizing surface
tension throughd
0. A derivation of this
Mullins–Sekerka instability was given in
Sec. 2.2.4.
It is natural to assume that this length
scale is related to the formation of side
branches. A direct estimate of the typical
wavelength
l
2of the side branches is
(2-92)
The remarkable result of the numerical
simulation is shown in Fig. 2-13. Appar-
ently, the ratio
l
2/l
Sis a constant of ap-
proximately 2.5, which is independent of
supersaturation and anisotropy. This seems
to be in quite good agreement with experi-
ments (Glicksman et al., 1976; Dougherty
et al., 1987; Honjo et al., 1985; Huang and
Glicksman, 1981).
The experimental comparison was made,
in fact, with an older theoretical concept
(Langer and Müller-Krumbhaar, 1978),
which did not correctly consider aniso-
tropy. By accident, however, the experi-
ll rs
20 22∂
S==ppld
mental anisotropy of the material succino- nitrile (Huang and Glicksman, 1981),
e≈0.1, gave the sames-value as the theo-
retical concept, and since
ecannot be var-
ied easily, there was no discrepancy.
To summarize these results, it appears
that the scaling relation, Eq. (2-92), shown in Fig. 2-13 from the numerical solution of the model in two dimensions, is in agree- ment with the experimental results.
We will now give a somewhat qualitative
explanation of the mechanism of side- branch formation as far as this can be de- duced from the theoretical approaches. A linear stability analysis (Langer and Mül- ler-Krumbhaar, 1978, 1980; Kessler and Levine, 1986a; Barber et al., 1987; Bar- bieri et al., 1987; Bensimon et al., 1987; Caroli et al., 1987; Kessler et al., 1987; Pelce and Calvin, 1987) indicates that the relevant modes for side-branch formation in the frame of reference moving with the tip consist of an almost periodic sinusoidal wave travelling from the tip down the shaft such that they are essentially stationary in the laboratory frame of reference (Langer and Müller-Krumbhaar, 1982; Deissler, 1987). The amplitude of these waves is not constant in space, but first grows exponen- tially in the tip region (Barbieri et al., 1987; Caroli et al., 1987). The exponential increase of that envelope in the tip region depends on the “wavelength” of the oscil- latory part (Bouissou et al., 1990).
In contrast to the earlier analysis by
Langer and Müller-Krumbhaar, all these modes are probably stable, so that without a triggering source of noise, they would decay, and a smooth needle crystal would result. Some driving force in the form of noise due to thermal or hydrodynamic fluctuations is needed to generate side branches, but apparently this is usually present. Estimates of the strength of these fluctuations (Langer and Müller-Krumb-
112 2 Solidification
Figure 2-13.Scaled sidebranch spacing l=l
2,nor-
malized with Ivantsov radiusR
0ands(e)
1/2
, plotted
versus capillary anisotropy for two supercoolings.
No dependence on
eorDis found, as expected (Saito
et al., 1988).www.iran-mavad.com
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165. 2.4 Free Dendritic Growth 113
haar, 1982; Barbieri et al., 1987; Langer,
1987a) are still somewhat speculative.
Given such a small noise at the tip, the
exponentially increasing envelope over the
side branches into the direction of the tail
then amplifies that noise so that the side
branches become visible. This happens
over a range of about two to ten side-
branch spacings. The actual selected wave-
length of the side branches in that tip re-
gion (assuming a white noise, triggering all
modes equivalently), according to these
considerations, is defined by the mode with
the largest amplitude at a distance of about
one “wavelength” away from the tip. This
is the product of the average amplitude due
to noise at the tip and the amplification fac-
tor from the envelope.
Langer (1987a) described the side-
branching deformation as a small (linear)
perturbation moving on a cylindrically
symmetric needle crystal (Ivantsov parabo-
loid). The noise-induced wave packets
generated in the tip region grow in ampli-
tude, spread and stretch as they move down
the sides of the dendrite producing a train
of side branches. In the linear approxima-
tion, the amplitude grows exponentially
and the exponent is proportional to|z|
1/4
.
These results are in approximate, qualita-
tive agreement with available experimental
observations (Huang and Glicksman, 1981;
Dougherty et al., 1987; Bisang and Bil-
gram, 1995), but experimentally observed
side branches have much larger amplitudes
than explicable by thermal noise in the
framework of the axisymmetric approach.
This means that either the thermal fluctua-
tion strength is not quite adequate to pro-
duce visible side-branching deformations,
or agreement with experiment would re-
quire at least one more order of magnitude
in the exponential amplification factor.
The description of the side-branching
problem, which takes into account the ac-
tual non-axisymmetric shape of the needle
crystal, defined by Eqs. (2-83a, b), was
given by Brener and Temkin (1995). They
found that the root-mean-squared ampli-
tude for the side branches generated by
thermal fluctuations is
(2-93)
where the functionx(z) is given by the
underlying “needle” solution, Eq. (2-83b),
and the fluctuation strengthQ
–
is given by
Langer (1987a),Q
–
2
=2k
BT
2
c
pD/(L
2
m
vr
4
).
The root-mean-squared amplitude for
the side branches increases with the dis-
tance from the tip,|z|. This amplitude
grows exponentially as a function of
(|z|
2/5
/s
1/2
). The important result is that
the amplitude of the side branches for the
anisotropic needle grows faster than for
the axisymmetric paraboloid shape. In
the latter casex(z)=2|z|
1/2
and the ampli-
tude grows exponentially as a function
of (|z|
1/4
/s
1/2
). This effect resolves the
puzzle that experimentally observed side
branches have much larger amplitudes than
can be explained by thermal noise in the
framework of the axisymmetric approach.
Agreement with experiment now is indeed
very good (Bisang and Bilgram, 1995).
Far down from the tip the side-branching
deformations grow out of the linear regime
and eventually start to behave like den-
drites themselves. It is clear that the
branches start to grow as free steady-state
dendrites only at distances from the tip
which are of the order of the diffusion
length which, in turn, is much larger than
the tip radius
rin the limit of smallP. This
means that there is a large range ofz,
1≤|z|≤1/P, where the side branches al-
ready grow in the strongly nonlinear re-
gime, but they do not yet behave as free
dendrites. We can think of some fractal ob-
〈〉
⎧
⎨
⎩
⎫
⎬
⎭x
s
1
212
32
12
2
33
( , ) ~ exp
/
/
/
ZY Q
x
z||www.iran-mavad.com
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166. ject where the length and thickness of the
dendrites and the distance between them
increase according to some power laws
with the distance|z|from the tip. The den-
drites in this object interact, owing to the
competition in the common diffusion field.
Some of them die and some continue to
grow in the direction prescribed by the an-
isotropy. This competition leads to the
coarsening of the structure in such a way
that the distance between the surviving
dendrites
l(z) is adjusted to be of the same
order of magnitude as the length of the den-
drites,l(z). The scaling arguments give
l(z)~l(z)~|z|(Brener and Temkin, 1995).
The morphology measurements on SCN
crystals yield a good quantitative agree-
ment with this linear law (Li and Becker-
mann, 1998).
We now summarize the presently estab-
lished findings for free dendritic growth
with respect to theirexperimental signifi-
cance. A discussion of additional effects
such as faceting will be given in Sec. 2.5.7
in the context of directional solidification.
For a given material with fixedD,d
0and
e, the growth rateVdepends upon super-
cooling
Dthrough Eq. (2-68), and with
Péclet numberPtaken from Eq. (2-56).
The dimensionless parameter
sis given in
Fig. 2-10. This is demonstrated for various
materials in Fig. 2-14. The size or tip ra-
dius of the dendrite is related to its velocity
by Eq. (2-90) and can be taken from Fig.
2-12. The typical wavelength of the side
branches is then given by Eq. (2-92) and
can be taken from Fig. 2-13. This provides
all the basic information that should be
valid in the tip region.
Beyond the understanding of steady-
state growth of the tip, the major new con-
cept that has emerged over the last few
years is that complex pattern formation
processes occurring on the much larger
scale of an entire dendrite grain structure
can be described by remarkably simple
“scaling laws”. The whole dendritic struc-
ture with side-branches looks like a fractal
object on a scale smaller than the diffusion
length and as a compact object on a scale
larger than the diffusion length (Brener et
al., 1996).
The new steady-state growth structures
that have been identified are the so-called
“doublons” in two dimensions (Ihle and
Müller-Krumbhaar, 1994; Ben Amar and
Brener, 1995), first observed in the form of
a doublet cellular structure in directional
solidification (Jamgotchian et al., 1993),
and the “triplon” in three dimensions (Abel
et al., 1997). Both structures have been
shown to exist without crystalline aniso-
tropy, unlike conventional dendrites. The
doublon has the form of a dendrite split
114 2 Solidification
Figure 2-14.Dimensionless growth rateV
˜
=Vd
0/2D
versus dimensionless undercooling
D. The scaling
quantity for the full curve (Langer and Müller-
Krumbhaar, 1977, 1978) was taken as
s= 0.025 (co-
incidentally in agreement with the anisotropy of suc-
cinonitrile). For references to the experimental points
see Langer (1980a). Excellent agreement between
theory (solid line) and experiment is found.www.iran-mavad.com
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167. 2.4 Free Dendritic Growth 115
into two parts about its central axis with a
narrow liquid groove between the two
parts, and triplons in three dimensions are
split into three parts. For a finite aniso-
tropy, however, these structures only exist
above a critical undercooling (or supersat-
uration for the isothermal solidification of
an alloy), such that standard dendrites
growing along·100Òdirections are indeed
the selected structures in weakly aniso-
tropy materials at low undercoolings, in
agreement with most experimental obser-
vations in organic and metallic systems.
From a broad perspective, the existence of
doublons and triplons is of fundamental
importance because it has provided a basis
on which to classify the wide range of pos-
sible growth morphologies that can form as
a function of undercooling and anisotropy
(Brener et al., 1996).
We have so far ignored the influence of
the kinetic coefficient
bin Eq. (2-88). This
omission is not likely to be very important
for low growth rates, but for fast growth
rates, as in directional solidification,
b
should be taken into account. We will re-
turn to this point in Sec. 2.5.
2.4.3 Experimental Results
on Free Dendritic Growth
The answer to the question of whether
dendritic growth is diffusion-controlled
or controlled by anisotropic attachment ki-
netics, was sought by Papapetrou (1935),
who was probably the first to make system-
aticin situexperiments on free dendritic
growth. He examined dendritic crystals
of transparent salts (KCl, NaCl, NH
4Br,
Pb(NO
3)
2, and others) under a microscope
in aqueous solutions and proposed that the
tip region should be close to a paraboloid
of rotational symmetry.
Many years later, the extensive and
systematic experiments by Glicksman and
his co-workers made an essential contribu-
tion to our understanding of dendritic
growth in pure undercooled melts (Glicks-
man et al., 1976; Huang and Glicksman,
1981). This research was initially con-
cerned mainly with highly purified succi-
nonitrile (SCN). It was extended to cyclo-
hexanol (Singh and Glicksman, 1989), wa-
ter (Fujioka, 1978; Tirmizi and Gill, 1989),
rare gases (Bilgram et al., 1989), and to
other pure substances with a crystal aniso-
tropy different from SCN such as pivalic
acid (PVA) (Glicksman and Singh, 1989).
Work on free growth of alloys includes
NH
4Cl–H
2O (Kahlweit, 1970; Chan et al.,
1978), NH
4Br–H
2O (Dougherty and Gol-
lub, 1988), SCN with acetone and argon
(Glicksman et al., 1988; Chopra et al.,
1988), PVA–ethanol (Dougherty, 1990),
and others.
The specific merit of the work of Glicks-
man et al. was that the systems for which
they characterized all the properties, in-
cluding surface energy, diffusion coeffi-
cient, phase diagram etc., have been exam-
ined. This led to clear evidence in the mid-
1970s that the theory of that time (using
extremum arguments for the operating
point of the tip) was not able to describe
the results quantitatively.
At the same time, Müller-Krumbhaar
and Langer worked on precisely the same
problem and proposed a theory based on
the stability of the growing dendrite tip,
called the marginal stability criterion (Lan-
ger and Müller-Krumbhaar, 1977, 1978).
Most of the existing experimental data
could be fitted using this criterion. Despite
the fact that this theory incorrectly ignored
the important role of anisotropy (as we
know now), it inspired a number of new
experiments and also attracted the interest
of other physicists.
As has been said before, today’s theory
is consistent with the older approximatewww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

168. theory if we allow for as(e) value that var-
ies with the anisotropy of the capillary
length. The corresponding central equation
for dendritic growth (Eq. (2) in Kurz and
Trivedi, 1990) should therefore still apply.
Pure substances(thermal dendrites)
Fig. 2-15 shows dendrites of two differ-
ent transparent materials with cubic crystal
structure: face-centered cubic PVA and
body-centered cubic SCN (Glicksman and
Singh, 1989). Qualitatively, both dendrites
look similar, but their branching behavior
shows some important differences. The un-
perturbed tip of PVA is longer, with a
sharper delineation of the crystallographic
orientations. Glicksman and Singh (1989)
found that PVA has a ten-fold larger sur-
face energy anisotropy than SCN (see Ta-
ble 2.1). The tip radii and growth rates as a
function of undercooling for both sub-
stances scale well when using the values
0.22 and 0.195 for
s, respectively (Fig. 2-
16). According to solvability theory, the
great difference in the anisotropy constant
eshould make a larger difference ins(e)
(compare with Fig. 2-10). The reason for
this discrepancy is not known, and we have
to leave this point to future research.
116 2 Solidification
Figure 2-15.Dendrite morphologies of two transparent materials with small melting entropies and cubic crys-
tal structures (plastic crystals); (a) pivalic acid (PVA) and (b) succinonitrile (SCN) (Glicksman and Singh,
1989).
(a) (b)www.iran-mavad.com
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169. 2.4 Free Dendritic Growth 117
behind the tip is delayed up to about seven
radii in PVA. This is quite consistent with
the recent calculations discussed in Sec.
2.5.4. The ratio of initial secondary arm
spacing
l
2over tip radiusRis also indi-
cated in Table 2-1.
Koss et al. (1999) have shown that even
in microgravity environment there is a
small but significant difference between
transport theory and experiment. Again
this has to be left to the future.
The secondary branch formation which
starts in SCN at a distance of three tip radii
Table 2-1.Experimentally determined dendrite tip quantities.
System Growth
s* R
2
V l
2/R d Reference
type [µm
3
/s]
Thermal dendrites Pure
Succinonitrile (SCN) Free 0.0195 3 0.005 Huang and
Glicksman (1981)
Pivalic acid (PVA) Free 0.022 7 0.05 Glicksman and
Singh (1986, 1989)
Cyclohexanol Free 0.027 Singh and
Glicksman (1989)
Solutal dendrites Alloy
NH
4Br–49 wt.% H
2O Free 0.081 ± 0.02 18 ± 3 5.2 0.016 ± 0.004 Dougherty and
Gollub (1988)
SCN–1.3 wt.% ACE Directional 0.032* 1300 2.1 ± 0.2 Esaka and
Kurz (1985)
SCN–4 wt.% ACE Directional 0.037* 441 ± 30 2.2 ± 0.3 Somboonsuk
et al. (1984)
CBr
4–7.9 wt.% C
2Cl
6Directional 0.044* 978 ± 8 3.18 Seetharaman
et al. (1988)
C
2Cl
6–89.5 wt.% CBr
4Directional 0.038* 124 ± 13 3.47 Seetharaman
et al. (1988)
Thermal and solutal Alloy
dendrites
SCN–ACE Free See reference Chopra et al.
(1988)
SCN–argon Free See reference Chopra et al.
(1988)
PVA–1 wt.% ethanol Free 0.05 ± 0.02 6 ± 1 0.006 ± 0.002 Dougherty
(1991)
PVA–2/4 vol.% ethanol Free 0.032 ± 0.006 35 6.8 Bouissou et al.
(1989)
* Due to differences in the definitions of
s* these values, as given in the corresponding literature, are smaller
by a factor of 2 with respect to the values used by Dougherty and Gollub (1988) and defined in this paper. The
values given here have been obtained by multiplying the original data by a factor of 2 in order to compare with
the same (one-sided) model. (See also Fig. 2-10.)www.iran-mavad.com
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170. Free alloy growth
(thermal and solutal dendrites)
In the free dendritic growth of alloys, an
interesting observation has been made by
various authors. For constant undercool-
ing, the growth rate first increases when
small amounts of a second substance are
added to a pure material, then reaches a
maximum, and finally drops and converges
with the pure solutal case. Early experi-
ments in this area by Fujioka and Linde-
meyer were first successfully analyzed by
Langer (1980c). Fig. 2-17 shows some re-
sults on SCN–ACE alloys from Chopra et
al. (1988). The increase inVis accompa-
nied by a decrease in the tip radius, which
sharpens due to the effect of solute. The
experimental findings can be compared
to two models: Karma and Langer (1984)
(broken line) and Lipton et al. (1987) (full
line). Both models provide at least qualita-
tively good predictions of the observed be-
havior. In their more recent calculations,
Ben Amar and Pelce (1989) concluded that
the simple model by Lipton et al. (1987)
is consistent with their more rigorous ap-
proach.
Table 2-1 gives a summary of represen-
tative results ofin situexperiments con-
cerning the dendrite tip.
Large undercoolings
Interesting experiments have also been
performed with pure and alloyed systems
under large driving forces, which reach
values beyond unit undercooling (for ex-
ample by Wu et al. (1987) and especially by
Herlach et al. (1990–1999), see Table 2-2).
Some of these are reproduced in Fig.
2-18 together with predictions from IMS
(Ivantsov–marginal stability) theory (Lip-
ton et al., 1987; Trivedi et al., 1987; Boet-
tinger et al., 1988) (with
s(e) = 0.025) and
including interface attachment kinetics.
118 2 Solidification
Figure 2-16.Effect of undercooling on (a) tip radius
and (b) growth rate of the two organic materials
shown in Fig. 2-15 (
a=D). The experimental results
superimpose as they are plotted with respect to dimen-
sionless parameters (Glicksman and Singh, 1989).
(a)
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171. 2.4 Free Dendritic Growth 119
These results are specifically of interest be-
cause two different techniques, one for the
determination of the diffusive speed and the
other for the undercooling–growth rate re-
lationship of the dendrites, have been cou-
pled in order to make the analysis as free
from adjustable parameters as possible. Up
to undercoolings of 200 K there is reason-
able agreement between experiment and
theory using the best fit for the measured
diffusive speed and the liquid diffusivity,
V
D= 26 m/s andD
L=2.7¥ 10
–9
m
2
/s (Ar-
nold et al., 1999). The lowerV(
∆T)-curve
is for a constant (equilibrium) value of the
distribution coefficientk(V)=k
eshowing
the importance of the appropriate velocity
dependence of the distribution coefficient.
At higher undercoolings other phenom-
ena take over.
One example is the grain refinement be-
yond a certain undercooling. This structure
has been explained by dendrite fragmenta-
tion due to morphological instability of the
fine dendrite trunks (Schwarz et al., 1990;
Karma, 1998).
Table 2-2.Dendritic growth velocity measurements in
highly undercooled melts; comparison between experi-
ment and theory (Herlach and coworkers, 1999–2000).
Metals and alloys
Co–Pd Volkmann et al. (1998)
Co–V Tournier et al. (1997)
Cu Li et al. (1996)
Ni Eckler and Herlach (1994)
Ni–Al Barth et al. (1994),
Assadi et al. (1998)
Ni–B Eckler et al. (1991a, 1992, 1994)
Ni–C Eckler et al. (1991b)
Ni–Si Cochrane et al. (1991)
Ni–Zr Schwarz et al. (1997),
Arnold et al. (1999)
Intermetallics
CoSi Barth et al. (1995)
FeAl Barth et al. (1997)
FeSi Barth et al. (1995)
Ni
3Al Assadi et al. (1996)
Ni
xSn
y Barth et al. (1997)
NiTi Barth et al. (1997)
Ni
xTi
yAl
z Barth et al. (1997)
Semiconductors
Ge Li et al. (1995a, 1996)
Ge–Cu Li and Herlach (1996)
Ge–Si Li et al. (1995b)
Ge–Sn Li and Herlach (1996)
Figure 2-17.Effect of dimensionless composition at constant undercooling of 0.5 K (2.1% of unit undercool-
ing) on (a) dimensionless growth rate and (b) on dimensionless tip radius for free dendritic growth in SCN–ace- tone alloys (Lipton et al., 1987). Points: experiments (Chopra et al., 1988); solid line: LGK model (Lipton et al., 1987); interrupted line: Karma and Langer (1984) model.
(a) (b)www.iran-mavad.com
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172. 2.5 Directional Solidification
Directional solidification is the most fre-
quent way in which a material changes its
state from solid to liquid. The necessary re-
moval of the latent heat of freezing usually
occurs in a direction prescribed by the lo-
cation of heat sinks: for a freezing lake, it
is the cold atmosphere above it, in casting
iron in a foundry, it is the cold sand mold,
into which the heat flow is directed.
At first, it may seem surprising to think
that anything interesting should happen at
the solid–liquid interface during this pro-
cess. In contrast to the situation described
earlier, in Secs. 2.4 and 2.2.4, the solid in a
casting process is cold and the liquid is hot,
so that we would expect the interface to be
stable against perturbations.
However, so far we have just considered
the solidification of a one-component ma-
terial, while in reality a mixture of materi-
als is almost always present, even if one of
the components is rather dilute. If, there-
fore, we assume that material diffusion is
the rate-determining (slow) mechanism,
while heat diffusion is much faster, the ori-
gin of a destabilization of the flat interface
can be easily understood on a qualitative
basis. We may consider one of the two
components of the liquid as an “impurity”,
which, instead of being fully incorporated
into the solid, is rejected at the interface.
Such excess impurities have to be diffused
away into the liquid in much the same way
as latent heat has to be carried away in the
case of a pure material as a rate-determin-
ing mechanism. Accordingly, precisely the
same destabilization and subsequent for-
mation of ripples and dendrites should oc-
cur.
Based on these qualitative arguments
we can expect the following modification
of the Mullins–Sekerka instability (Sec.
2.2.4) to occur in the present situation of
directional solidification. The diffusion of
material together with capillary effects pro-
duces a spectrum for the growth rates or the
decay rates similar to Eq. (2-50), while the
temperature field acts as a stabilizer, inde-
pendent of the curvature of the interface,
when a constant term (independent ofV)
inside the brackets of Eq. (2-50) is sub-
120 2 Solidification
Figure 2-18.Dependence on total bath undercooling
of Ni–1 wt.% Zr alloy. (a) Dendrite growth velocity,
as measured (dots), and results from Ivantsov–mar-
ginal stability dendrite growth theory using the val-
uesV
D= 26 m/s andD
l=2.7¥ 10
–9
m
2
/s. The lower
curve in (a) is for local equilibrium partition. (b) The
calculated dendrite tip radius, (c) the computed inter-
face compositions and (d) the individual undercool-
ing contributions: thermal undercoolingDT
t, consti-
tutional undercoolingDT
c, curvature undercooling
DT
r, and attachment kinetic undercoolingDT
k(Ar-
nold et al., 1999).www.iran-mavad.com
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173. 2.5 Directional Solidification 121
tracted. At low solidification rates, the flat
interface is stable; above a critical speed, it
becomes unstable against the formation of
ripples, cells and dendrites.
In the next section, a few thermody-
namic questions related to interface prop-
erties in two-component systems are con-
sidered, before describing patterns in direc-
tional solidification.
2.5.1 Thermodynamics
of Two-Component Systems
There is a vast amount of literature avail-
able on the thermodynamics of solidifica-
tion and on multi-component systems (for
example Callen, 1960; Baker and Cahn,
1971). Despite this fact, to further the clar-
ity of presentation, we would like to at least
sketch the tools that may be used to gener-
alize some approximations which will be
made in the next sections.
The fundamental law of thermodynam-
ics defines entropy as a total differential in
relation to energy and work:
dU=TdS–PdV+
S
i
m
idN
i (2-94)
with energyU, entropyS, volumeV, pres-
sureP, particle numbersN
ifor each species
and chemical potential
m
i. The energy is a
homogeneous function of the extensive
variables
(2-95)
U(bS,bV,bN
i,…)=bU(S,V,N
i,…)
with an arbitrary scale parameterb>0.
With the differentiation rule d(XY )=
XdY+YdX, other thermodynamic poten-
tialsU
˜
are obtained fromUby Legendre
transformations
U
˜
=U–
S
j
X
jY
j (2-96)
whereX
jare some extensive variables, and
Y
jthe corresponding intensive variables.
The Helmholtz energyFis then
F(T,V,N
i,…)=U–TS;
dF=–SdT–PdV+
S
i
m
idN
i (2-97)
and the often-used Gibbs energyGis
G(T,P,N
i,…)=U –TS+PV= S
i
m
idN
i;
dG=–SdT+VdP+
S
i
m
idN
i (2-98)
At atmospheric pressure in metallurgical
applications, the differences betweenFand
Goften can even be neglected, but gener-
ally the Gibbs energy, Eq. (2-98), is most
frequently used. It follows immediately that
the chemical potentials
m
iare defined as
(2-99)
The thermodynamic equilibrium for a sys-
tem is defined by the minimum of the re-
spective thermodynamic potential with re-
spect to all unconstrained internal parame-
ters of the system. If the system consists
of two subsystemsaandbin contact with
each other, then in thermal equilibrium the
temperatures, pressures, and the chemical
potentials for each particle typeimust be
equal:
T
a=T
b,P
a=P
b,m
i,a=m
i,b (2-100)
For the case under consideration we have a
solid phasea(with assumed low concen-
tration of B atoms) and a liquid phaseb
(with higher concentration of B atoms).
For simplicity, we further assume that the
atomic volumes of both species are the
same and unchanged under the solid–liq-
uid transformation.
For a system with a curved interface
between a solid and a liquid of different
compositions, the chemical potentials can
be calculated as follows. Assuming thatN
A
particles of solvent andN
Bparticles of
solute are given, there will be an unknown
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
G
N
i
TP
i
,
=mwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

174. number ofN
Sparticles in the solid andN
L
particles in the liquid, whose composition
is still undetermined. Define the numbern
as the number of B particles in the solid.
KeepingN
Sandninitially fixed, the Gibbs
potential is obtained as
G(N
A,N
B,N
S,n)=N
Sg
S(C
S)+N
Lg
L(C
L)
+4pR
2
g (2-101)
where
gis the surface free energy density,
Ris the radius of the solid sphere
N
S=–
4
3
pR
3
(unit atomic volume), andg
S
andg
Lare free energy densities for homo-
geneous solids or liquids at concentrations
C
SandC
L. Removing the constraints onN
S
andn, we obtain thermodynamic equilib-
rium by minimizingGwith respect toN
S
andn,sothatG =G(N
A,N
B). Together
with the chemical potentials from Eq. (2-
99), this gives
(2-102)
with curvatureK=2/Rand
m=m
B–m
Abe-
ing the difference in chemical potentials
between solute and solvent. The values
m
0,
C
0
L
,C
0
S
correspond to equilibrium atg=0
as a reference, around whichg(C) was lin-
earized. Here
gwas assumed to be inde-
pendent of curvature and concentrations,
but this can easily be incorporated (for ef-
fects of surface segregation, for example).
This equation is the boundary condition
for the chemical potential on the surface of
a solid sphere of surface tension
gcoexist-
ing with a surrounding liquid of higher
concentrationC
Lof B atoms. From a prac-
tical point of view, the formation in terms
of chemical potentials does not look very
convenient, as they are not directly mea-
surable. From a theoretical point of view,
this has advantages, as the chemical poten-
tial is the generalized force controlling
matter flow and phase changes. In particu-
mm
g−−
−
0 00=
LS()CC
K
lar, the spatial continuity of chemical po- tentials together with continuous tempera- ture and pressure guarantees local thermal equilibrium, which we will assume to hold in most of the following discussions.
We now come to the discussion of a typ-
ical phase diagram for a two-component system (Fig. 2-19). The vertical axis de- notes the temperature, the horizontal axis the relative concentration of “solute” in a “solvent”, or, more generally, B atoms rel- ative to A atoms.
At high temperatures,T>T
0, the system
is liquid, regardless of concentration. As- suming a concentrationC
•to be given in-
itially, we lower the temperature toT
1.At
122 2 Solidification
Figure 2-19.Typical solid–liquid phase diagram for
a two-component system with the possibility of eu-
tectic growth.C
˜
L,C
˜
Sare the liquidus and solidus
lines with a solid–liquid two-phase region in
between. For a relatively low concentrationC
•of so-
lute in the liquid, a single solid phase at the same
compositionC
S
0=C
•can show stationary growth. In
directional solidification, a positive temperature gra-
dient∂T/∂zis assumed to be given perpendicular to
the solid–liquid interface and to advance in +z-di-
rection. The advancing interface chooses its position
to be at temperatureT
0, the concentration in the liq-
uid at the interface is then atC
L
0. Ahead of the inter-
face (z= 0), the concentration profileC(z)decays
towardsC
•atz=•(dashed curve). As long as
C(z)>C
˜
L(T(z)) withT(z)=T
0+z∂T/∂z, a flat inter-
face remains stable.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

175. 2.5 Directional Solidification 123
this temperature, we first hit the liquidus
lineC
˜
L(T), and the system begins to solid-
ify, producing a solid of very low concen-
tration marked by the solidus lineC
˜
S(T).
When the temperature is slowly lowered,
solidification becomes complete atT
0.At
lower temperatures, the whole system is
solid.
The region betweenC
˜
S(T)andC
˜
L(T)is
the two-phase region: if we prepare a
system at a concentration betweenC
0
S
and
C
0
L
at high temperatures and then quickly
quench it toT
0, the system starts to sepa-
rate into a solid phase of concentrationC
0
S
and a liquid phase ofC
0
L
. In practice, this is
a very slow process, with lengths varying
with timetapproximately ast
1/3
(Lifshitz
and Slyozov, 1961; Wagner, 1961).
In the case of directional solidification, a
thermal gradient in the system defines a di-
rection such that the liquid is hot and the
solid is cold. A flat interface may then be
present at a position in space at tempera-
tureT
0. For equilibrium between solid and
liquid at that temperature, the concentra-
tion in the solid must be atC
0
S
=C
•, and in
the liquid atC
0
L
=C
˜
L(T
0). We now assume
that the liquid at infinity has concentration
C
•. Clearly there must be a decrease in
concentration as we proceed from the inter-
face into the liquid. In order to maintain
such an inhomogeneous concentration, the
interface must move toward the liquid.
In other words, when the liquid of com-
positionC
0
L
freezes, the solid will only
have a concentrationC
0
S
. The difference in
concentrations
DC=C
0
L
–C
0
S
(2-103)
is not incorporated but is pushed forward
by the advancing interface and must be car-
ried away through diffusion into the liquid.
This is equivalent to the latent heat gener-
ated by a pure freezing solid, discussed in
Sec. 2.2.3. Therefore, we expect a spatial
concentration profile ahead of the interface
which decays exponentially fromC
0
L
at the
interface toC
•far away from the interface.
But why should it decay toC
•(or why
should the interface choose a temperature
position such thatC
0
S
=C
•)?
The answer is quite simple, and again
was given in similar form in Sec. 2.2.3, Eq.
(2-33) for the pure thermal case: ifC
0
S
were
not identical toC
•, then during the solid-
ification process there would be either a to-
tal increase (or decrease) of concentration
– which clearly is impossible – or at least
the concentration profile could not be sta-
tionary.
This is a rather strict condition, which
we can reformulate as follows: if we im-
pose a fixed temperature gradient∂T/∂z
and move this at fixed speedV
0over a
system of concentrationC
•at infinity in
the liquid (toward the liquid in the positive
z-direction), then the interface will choose
a position such that its temperature is atT
0,
the concentration in the solid will beC
0
S
(averaged parallel to the interface), and the
liquid concentration at a flat interface will
be atC
0
L
. This follows from global conser-
vation of matter together with the imposed
stationary solidification rate.
As a final point, we can even derive a
condition for the stability of the interface.
The concentration profile in the liquid will
decay exponentially with distancezaway
from the interface as
C
L(z)=(C
0
L
–C
•)e
–2z/l
+C
• (2-104)
by analogy to Eq. (2-32). Since we assume
the temperature gradient
G
T=∂T/∂z> 0 (2-105)
to be fixed, the temperature varies linearly
with distancezfrom the interface. This may
be written (byG
T=(T–T
0)/z)as
(2-106)
C
L(T)=(C
0
L
–C
•)e
–2(T–T
0
)/(lG
T
)
+C
•www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

176. and incorporated into Fig. 2-19 as a
dashed-dotted line. Note that the diffusion
lengthlis again defined asl=2D/V
0, with
Dbeing the diffusion constant of solute
atoms in the solvent, andV
0the interface
velocity imposed by the advancement rate
of the temperature gradient.
From Eq. (2-106), it is obvious that the
dashed-dotted concentration line in Fig. 2-
19 converges very quickly toC
•for high
solidification speedsV
0. As long as that
concentration line is fully in the liquid re-
gion of the phase diagram, nothing specific
happens. But if the dashed-dotted line
partly goes through the two-phase region
betweenC
˜
S(T)andC
˜
L(T), the liquid in
front of the interface is supercooled! This
implies the possibility of an instability of
the solid–liquid interface, which is com-
pletely analogous to the discussion in Sec.
2.2.4.
A sufficient condition for stability of the
interface in directional solidification is
therefore
(2-107)
so that the dashed-dotted curve remains
outside the two-phase region (Mullins and
Sekerka, 1963; Langer, 1980a). Here we
have assumed that material diffusion in the
solid can be ignored. In fact, in practical
situations, violation of this condition typi-
cally means “instability” of the interface,
so that cellular or dendritic patterns are
formed. The reason for this latter conclu-
sion is that the effect of stabilization due to
capillarity (or surface tension) is rather
weak for typical experiments at threshold.
In summary, in this section we have
derived both the boundary condition – in
terms of chemical potential – for a curved
interface and a basic criterion for interface
stability during directional solidification.
DC
V
DG
C
T
T
<
d
d
L
˜
2.5.2 Scaled Model Equations
A theoretical analysis of practical situa-
tions of directional solidification suffers – among other problems – from the many rel- evant parameters entering the description. The usual way to proceed in such cases is to scale out as many parameters as pos- sible, writing the problem in dimensionless variables. We have done this already in the discussion of free dendritic growth by in- troducing the dimensionless diffusion field u; in hydrodynamic applications, it is com-
mon practice to use Reynolds and Rayleigh numbers (Chandrasekhar, 1961).
For our present problem, we will pro-
ceed in an analogous way. The first step is to express all experimental parameters (wherever possible) in length units (i.e., diffusion length, capillary length, etc.). For presenting results, we divide these lengths by the thermal length introduced below, as this is a macroscopic length which will ap- proximately set the scale at the onset of the instability.
Directional solidification involves chem-
ical diffusion of material together with heat diffusion. As heat diffusion is usually faster by several orders of magnitude, we may often assume constant temperature gradients to exist in the liquid and in the solid. Furthermore, we will also assume that there are equal thermal diffusivities in liquid and solid, which is often the case within a few percent, but this has to be checked for concrete applications. The dif- fusion field to be treated dynamically then corresponds to the chemical concentration.
It is clear from the discussion in the pre-
vious section that for a flat interface mov- ing at constant speed there is a concentra- tion jumpDC=C
0
L
–C
0
S
across the inter-
face, while in the liquid and in the solid, the termC
•is approached asymptotically
because of the condition of stationary
124 2 Solidificationwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

177. 2.5 Directional Solidification 125
movement, together with the global con-
servation of matter. We therefore normalize
the diffusion field in the liquid to
(2-108)
so that it varies from one to zero in the pos-
itivez-direction from the interface atz=0
toz=•. If the interface is not at position
z= 0 but at
z, we must requireu
L=1–z/l
T,
because at a distance
(2-109)
the liquidus concentration has reached the
asymptotic value. This is the thermal
length which we assume to be fixed by the
thermal gradientG
T, the concentration
jumpDC, and the liquidus lineC
˜
(T),
which is here assumed to be a straight line
in theTvs.Cdiagram.
The equation of motion in quasi-station-
ary approximation then becomes, in anal-
ogy to Eq. (2-52) withl=2D
L/V
(2-110)
This equation applies equivalently to the
solid but with a different chemical diffu-
sion lengthl¢due to different chemical
diffusion constants. The boundary condi-
tion in analogy to Eq. (2-88) obviously be-
comes
u
L(z)=1–dK – z(x,t)/l
T–bV
^(2-111)
where we now have
D=1 as the first term
on the r.h.s., with curvatureKbeing posi-
tive for the tip of a solid nose pointing into
the liquid. The capillary lengthdis dis-
cussed below and interface kinetics with
b≠0 will be discussed in Sec. 2.5.4. The
solid boundary condition is simply
u
S=k(u
L– 1) (2-112)
1
0
2
2
D
uu
l
u
z
t
L
LL
L =∂≈∇ +
∂
∂
l
C
G
T
C
T
T=
d
d
L
D
˜
u
CC
C
L=
(Liquid)−
∞
D
with segregation coefficientk(equilibrium
values assumed) defined as
(2-113)
through the slopes of the liquidus and sol-
idus lines. When they intersect at {T
m,
C= 0} this is equivalent to the conventional
definitionk=C
S/C
L, but in the above for-
mulation,k=1 may also be true for a con-
stant jump in concentrations, independent
of temperature.
The conservation law at the interface
z=
zfinally becomes
V
^{1 + (1 –k)(u
L– 1)}
=–D
Lnˆ·—u
L+D
Snˆ·—u
S (2-114)
whereV
^is the interface velocity in direc-
tionnˆnormal to the interface. Fork=1, the
brackets {…} give 1, corresponding to a
constant concentration jump, while for
k= 0, they giveu
L, since for a solid in Eq.
(2-112),u
S=0.
This standard model for directional so-
lidification (Saito et al., 1989) therefore
consists of Eqs. (2-108) to (2-114), to-
gether with an additional diffusion equa-
tion as Eq. (2-110) inside the solid phase.
The open point is finally the relation of
the capillary lengthd(Eq. (2-111)) with ex-
perimentally measurable material parame-
ters. As a first step, we interpret theu-field,
Eq. (2-108), as a scaled form of the chemi-
cal potential
m(see Sec. 2.5.1) nearT
0
(2-115)
assuming that linearizing
maround its equi-
librium value at the liquidus lineC
˜
L(T
0)is
sufficient to describe its dependence upon
C. By the definitions in Eq. (2-102) to-
gether with Eq. (2-111), we now obtain the
capillary length in the form given in Eq.
(2-86) for the chemical case. Here we have
u
CC
L
L=
mm
m−
∂∂
∞
D(/
˜
)
k
T
C
T
C
=
d
d
d
d
LS
˜˜www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

178. generalized to anisotropicgas derived
in Eq. (2-21). It can finally be related to
measurable quantities using the Clausius–
Clapeyron equation for the latent heatL
m
of freezing of a solution atT
m
L
m=–T
mDC(dm/dT)
coex (2-116)
where (d
m/dT)
coexis the slope of the coex-
istence line when
mis plotted againstT.
Together with
(2-117)
this gives for the chemical capillary length,
d, in the limit of smallDC
(2-118)
Certain approximations used here, such as
the linearization involved in Eq. (2-115) or
neglecting of∂
m/∂Tin Eqs. (2-118), may
not be safe for the case of a large segrega-
tion coefficientk≈1orwhenDC is large
(Langer, 1980a). In most practical applica-
tions, however, this is a minor source of er-
ror in comparison with other experimental
uncertainties. Furthermore, the concentra-
tion jumpDCin Eq. (2-118) is kept fixed,
while in reality it should correspond to the
temperature-dependent difference in con-
centration between the liquidus and solidus
lines. Both for slow and fast growth rates,
however, this only gives a minor correc-
tion, and we will ignore its effect in order
to facilitate comparison with free dendritic
growth.
In summary, with this model we now
have all the ingredients to discuss some ba-
sic features of directional solidification by
analytical and numerical tools. The presen-
tation in the scaled form may not seem at
first to be the most convenient means of di-
rect comparison with experiments. Its great
advantage over an explicit incorporation of
d
T
CL T C
=
m
mL
[()]
/
˜gg+′′
∂∂J
D| |
d
d
=
d
d
coex
L
mm m
TC
C
TT
⎛
⎝
⎞
⎠
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟ +
∂
∂
˜
all parameters is that qualitatively different
behavior always corresponds to different
ratios of length scales or time scales rather
than some differences in absolute mea-
sures, and, consequently, this presentation
allows for a more intuitive formulation of
results.
2.5.3 Cellular Growth
A plane interface between the solid
phase and the liquid phase of a two-compo-
nent system tries to locate its position in a
thermal gradient so that the chemical po-
tentials of both components are continuous
across the interface. Under stationary
growth conditions, that position corre-
sponds to a temperature, such that the con-
centration in the solid (solidus line of the
phase diagram) is equal to the concentra-
tion in the distant liquid. This growth mode
persists for velocities up to a critical veloc-
ity, above which the interface undergoes a
Mullins–Sekerka instability toward cel-
lular structures. A necessary condition for
this instability to occur follows from Eq.
(2-107), which can be written in terms of
chemical diffusion lengthl=2D/Vand
thermal diffusion lengthl
T, Eq. (2-109), as
l/l
T⎡2 (2-119)
The 2 comes from the specific definition of
l, and the inequality for instability is only
approximate because minor surface tension
effects have not yet been considered here.
Incorporation of surface tension reveals
that the instability first occurs for a critical
wavelength
l
clarger than the stability
length
l
s=2p÷
--
dl. Slightly above the criti-
cal pulling speedV
c, the interface makes
periodic structures of finite amplitude. This
was analyzed by Wollkind and Segel
(1970), and for other specific cases, by
Langer and Turski (1977). A more general
treatment was given by Caroli et al. (1982).
126 2 Solidificationwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

179. 2.5 Directional Solidification 127
The result of these investigations is that
in a diagram of pulling speedVversus
wavelength
lthere exists a closed curve of
neutral stability, Fig. 2-20. At fixedV,a
small amplitude perturbation of the inter-
face at a wavelength on that curve neither
grows nor decays. Perturbations at wave-
lengths outside that curve decay, inside the
curve they grow to some finite amplitude.
This is similar to periodic roll patterns in
the Rayleigh–Benard system of a fluid
heated from below (Chandrasekhar, 1961),
but here a maximal speed,V
a, is present,
above which a flat interface is absolutely
stable. For normal alloys, this speed is very
high, while for liquid crystals, it is more
easily accessible in controlled experiments
(Bechhoefer et al., 1989). The diffusion
length atV
ais of the order of the capillary
length.
At low speeds, but slightly above the
critical velocity,V
csinusoidal “cells” will
be formed for systems with segregation co-
efficientsknear unity. For small segrega-
tion coefficients, however, the neutral
curve does not define such a normal bifur-
cation but rather an inverse bifurcation.
This means that immediately aboveV
c
large amplitude cells with deep grooves are
formed. A time sequence of the evolution
of a sinusoidal perturbation into elongated
cells at 1% aboveV
cdue to inverse bifurca-
tion is shown in Fig. 2-21. This can be
understood theoretically (Wollkind and
Segel, 1970; Langer and Turski, 1977; Ca-
roli et al., 1982) by means of an amplitude
equation valid nearV
c:
(2-120)
whereAis the (possibly complex) ampli-
tude of a periodic structure exp (ikx) with
k=2p/l
c. The coefficienta
1is called the
Landau coefficient. If it is positive, we have
∂
∂
−⎛
⎝
⎜
⎞
⎠
⎟−+…
A
t
VV
V
AaAA=
c
c
1
2
||
Figure 2-20.Neutral stability curve for a flat solid–
liquid interface in directional solidification (sche-
matic). The dependence of the growth rate (pulling
speed)Von the wavelength
lof the interface pertur-
bation is approximatelyV~
l
–2
, both for the short
and long wavelength part of the neutral stability
curve,V
candV
aare the lower critical velocity and
upper limit of absolute stability, respectively.
Figure 2-21.Time evolution of an interface from si-
nusoidal to cellular structure slightly above the criti- cal thresholdV
cfor the case of inverse bifurcation. A
secondary instability quickly leads to a halving of the wavelength.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

180. a normal bifurcation with|A|~÷
-----
V–V
c,
while fora
1< 0, the third order term does
not stabilize the pattern but allows very
large amplitudes leading to elongated cells
(Fig. 2-21), which will be stabilized by
some higher-order effects.
A second phenomenon is usually asso-
ciated with this inverse bifurcation, namely,
the splitting of the wavelength
l
cÆl
c/2.
Qualitatively, this is understandable from
nonlinear corrections since the squaring of
the original pattern ~ exp (ikx) produces
terms ~ exp (i 2kx). This effect has clearly
been observed in experiments (de Che-
veigne et al., 1986).
We will discuss some aspects of the very
high speed region in Sec. 2.5.7 but devote
the main part of the following discussions
to the most interesting region for practical
purposes, which is not too close to the
upper and lower bounds of the growth rate
V
aandV
c.
Approximating by straight lines the neu-
tral curve of the logarithmic plot Fig. 2-20
in the intermediate velocity region, we find
for both the small and the large
llimits the
relation
V
l
2
≈constant (2-121)
Again we have recovered the form of Eq.
(2-1) mentioned in the introduction as a
scaling law where
lhere is the cell spac-
ing. This suggests that the cellular pattern
formed in actual experiments would also
follow this behavior. Unfortunately, this
problem has not yet been settled to a satis-
factory degree from a theoretical point of
view. This is partly due to the difficulty of
finding good analytical approximations to
the cellular structures, which makes nu-
merical calculations necessary to a large
degree. We will return to this point in Sec.
2.5.5.
For small amplitude cells obtainable
under normal bifurcation, some progress
has been made (Brattkus and Misbah,
1990). A phase-diffusion equation has been
derived describing the temporal evolution
of a pattern without complete periodic vari-
ation of the interface. The basic idea is to
replace the periodic trial form exp (ikx)
by a form exp (i
Q(x,t)) so thatq(x,t)=
∂
Q/∂xis now no longer a constant but is
slowly varying in space along the interface
and evolving with time. We can derive a
nonlinear phase diffusion equation
∂
tq=∂
x{D
˜
(q)∂
xq} (2-122)
with a diffusion coefficientD
˜
(q) depend-
ing in a complicated way onq. The proce-
dure is well known in hydrodynamics and
it is associated there with the so-called
Eckhaus instabilility (Eckhaus, 1965). This
instability eventually causes an (almost)
periodic spatial structure to lose or gain
one “period”, thereby slightly changing the
average wavelength. In directional solidifi-
cation, the result (Brattkus and Misbah,
1990) is shown in Fig. 2-22, where velocity
is plotted against wavenumber in a small
interval above the critical velocity. The full
line is the neutral curve, the full triangles
mark the Eckhaus boundary of phase
stability. A periodic (sinusoidal) pattern is
stable against phase slips only inside the
region surrounded by triangles, thereby al-
lowing for an Eckhaus band of stationary
periodic solutions with a substantially re-
duced spread in wavenumbers as compared
to the linear stability results. Note also that
the results for phase stability (dashed line)
from the amplitude equation only hold in
an extremely small region aboveV
c, while
only 20% aboveV
c, it shows no overlap
with the result from the present analysis
(triangles). The short-wavelength branch
has a very complicated structure, while the
long-wavelength branch far from the
threshold scales as
l
Eck~V
–1/2
, again like
the neutral curve. This also seems to be in
128 2 Solidificationwww.iran-mavad.com
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181. 2.5 Directional Solidification 129
agreement with experimentally observed
results, as discussed later (Billia et al.,
1987, 1989; Somboonsuk et al., 1984;
Esaka and Kurz, 1985; Eshelman and Tri-
vedi, 1987; Faivre et al., 1989; Kurowsky,
1990).
At higher velocities, the cells quickly be-
come elongated (Fig. 2-23) with deep
grooves forming bubbles. This was first
obtained through numerical calculations by
Ungar and Brown (1984a, b, 1985a, b).
Calculations with a dynamical code in
quasi-stationary approximation (Saito et
al., 1989) confirmed the stability of these
structures with respect to local deforma-
tions and short-wavelength perturbations.
The long-wavelength Eckhaus stability has
not been investigated yet for these cells.
All calculations were made in two dimen-
sions, which are believed to be appropriate
for experiments of directional solidifica-
tion in a narrow gap between glass plates.
At higher velocities and wavelengths (or
cell sizes) not much smaller than the dif-
fusion length, the grooves become very
narrow, similar to Fig. 2-21 (Ungar and
Brown, 1984a, b, 1985a, b, Karma, 1986,
Kessler and Levine, 1989, McFadden and
Coriell, 1984, Pelce and Pumir, 1985).
Figure 2-22.Stability diagramVvs. 2p/ lnear the lower critical threshold for a flat moving interface in direc-
tional solidification. The solid line is the neutral stability curve Fig. 2-20, the dotted curve is the most danger-
ous mode, the dashed curve is the limit of the Eckhaus stability from the amplitude equation. The triangles mark
the Eckhaus limit as obtained from the full nonlinear analysis (Brattkus and Misbah, 1990), with stable cellular
interfaces possible only inside that region. The band width of possible wavelengths for cellular interfaces ac-
cordingly is smaller by a factor of≈0.4, as compared to the band width given by the neutral (linear) stability
curve.
Figure 2-23.Computed example for a deep cellular
interface atV≈5V
cwith bubble formation at the bot-
tom of the groove (Saito et al., 1990).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

182. If the velocity is fixed and the wavelength
lis reduced significantly below the diffu-
sion lengthl, the Saffmann–Taylor limit is
reached (Brener et al., 1988; Dombre and
Hakim, 1987; Kessler and Levine, 1986c),
which is equivalent to a low-viscosity fluid
being pushed into a channel of width
l
filled with a high-viscosity fluid. The low-
viscosity fluid forms a finger just like the
solid in directional solidification. Near the
tip, the width of the finger
l
fcorresponds to
l
f=Dl
with cell spacingl, whereD<1 is the ac-
tual supercooling at the tip (remember that
D=1 for a flat interface atz= 0, andD=0
for a flat interface at
z=l
T). This serves
to verify the consistency of numerical
calculations (Saito et al., 1989). An even
more detailed analysis was carried out by
Mashaal et al. (1990).
For comparison with experiments, it is
useful to draw aVvs.
ldiagram (Fig. 2-
24). Here the full line is again the neutral
curve, the broken line is the most danger-
ous (or most unstable) mode, and the dot-
ted line denotes the relationl=
l, where the
diffusion length is equal to the imposed
wavelength. The asterisks mark some de-
tailed numerical investigations (Saito et al.,
1989). The asterisk furthest to the left is
close to the above-mentioned Saffmann–
Taylor limit. At slightly higher wave-
lengths, where
l<lstill holds, we are in a
scaling region, where the radius of curva-
ture at the tips of the cells is about 1/5 of
the cell spacing, as also found experimen-
tally (Kurowsky, 1990). All these consider-
ations give sufficient confidence that the
numerical calculations may also provide
insight into the mechanism of directional
solidification for the most interesting case
130 2 Solidification
Figure 2-24.“Phase”-diagram log (V)vs.log( l) for interface patterns in directional solidification. The solid
and dashed curves denote the neutral stability curve, and the dashed-dotted curve the most dangerous mode.
Asterisks mark fixed parameter values discussed hereafter. The lower critical threshold here isV
c≈1,l
c≈0.5 for
velocity and cell spacing. For other parameters see text. At low pulling speeds and high wavelengths cellular
patterns with narrow grooves are found (a). At very short wavelengths and moderate speeds cellular patterns
with wide grooves are found, consistent with theories for viscous fingering. At high pulling speeds, such that
the cell spacing
lis significantly wider than the diffusion lengthl, side-branching dendritic arrays are formed
(c) (Saito et al., 1990). The speeds are still much smaller than the absolute stability limitV
a.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

183. 2.5 Directional Solidification 131
of dendritic arrays formed at higher growth
rates.
A few words on numerical methods and
system parameters may be in order before
we discuss the dendritic region. The nu-
merical code is equivalent to the one used
for the free dendritic case with the modifi-
cation that it is necessary to integrate over
several cells to arrive safely outside the
diffusion length. Furthermore, in principle,
diffusion has to be considered in both the
liquid and the solid. Since the diffusion co-
efficient for material in the solid is usually
much lower than it is in the liquid, it is
found that diffusion in the solid alloy can
usually be neglected on time scales for the
formation of cells. For long durations, of
course, microsegregation takes place, and
solid diffusion then becomes important
(Kurz and Fisher, 1998).
A more serious difficulty in directional
solidification is the large number of param-
eters defining the system. We will concen-
trate here on typical parameter values used
in experiments performed for some trans-
parent materials between glass plates. Sev-
eral tests and specific calculations also
done for alloys, however, indicate that a
large part of the results can be carried over
to these more relevant situations from a
metallurgical point of view without qual-
itative changes.
2.5.4 Directional Dendritic Growth
The diagram in Fig. 2-24 showing veloc-
ity versus
lin logarithmic form indicates
that qualitatively different behavior may be
expected depending on whether the diffu-
sion length is larger or smaller than the cell
spacing. In the previous section, we dis-
cussed the first case. When the diffusion
length becomes smaller than the cell spac-
ing we expect that the individual cells be-
come more and more independent of each
other, until finally they may behave like in-
dividual isolated dendrites.
In order to test this hypothesis, a series
of numerical experiments were performed
at a fixed cell spacing and increasing pull-
ing velocity (Saito et al., 1989). The nu-
merical parameters of the model were rep-
resentatively taken to correspond to steel
with Cr–Ni ingredients (Lesoult, 1980). In
dimensionless units, the critical velocity
and wavelength for the plane-front in-
stability wereV
c= 1.136,l
c= 0.514. The
anisotropy of the capillarity length was not
known and was taken as
e= 0.1 to allow for
comparison with the previously mentioned
calculations on the free dendritic case. The
cellular wavelength was fixed to
l= 0.36,
corresponding to the asterisks at increasing
velocity and constant
lin Fig. 2-24.
At the lowest velocity still below the
l=
ldividing line, rounded cells were ob-
served; the tip was not well approximated
by a parabola. At higher speeds (V=12) the
parabolic structure of the tip became vis-
ible, Fig. 2-25, and at even higher speeds
(V= 20) the dendritic structure with side
branches was fully developed, Fig. 2-26.
We can now compare the resulting tip ra-
dius with the predictions from free den-
dritic growth. Note that in the present case
the velocity is fixed rather than the super-
cooling, so that the dendrite now uses a
supercooling corresponding to the given
velocity. This means that the tip of the den-
drite is no longer at a position in the tem-
perature-gradient field like a flat interface,
but has advanced toward the warmer liq-
uid.
Fig. 2-27 contains the ratio of the tip ra-
dius divided by the radius from scaling, Eq.
(2-68) (where the Péclet numberPwas
used in the original form as the ratio of tip
radius to diffusion length). Furthermore,
this figure gives the ratio of the tip radius
to the Ivantsov radius, which comes fromwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

184. the Péclet number through the relation for
the supercooling at the tip, Eq. (2-56).
The data are instantaneous measurements
rather than time-averaged measurements.
It is obvious from Fig. 2-27 that the scaling
relation, Eq. (2-68), holds very well at
rather low speeds, where neighboring cells
still interact substantially through the dif-
fusion field, while the relation from the
Ivantsov formula for the Péclet number
only holds at higher velocities. The obvious
reason for the latter deviation at low veloc-
ities is that the Ivantsov relation represents
a global conservation law for an isolated
parabolic structure, which is clearly not
valid when several cells are within a diffu-
sion length.
The observation that the scaling relation,
Eq. (2-68), is very robust obviously has to
do with the fact that it results from a solv-
ability condition at the tip of the dendrite,
which is only very weakly influenced by
deformations further down the shaft.
In the same study, it was also confirmed
that the side branches fulfilled precisely
the same scaling relation (Fig. 2-13) as the
132 2 Solidification
Figure 2-25.Transition from needle-shaped to dendritic cells at increasing pulling speeds.V= 4 is below the
dotted line in Fig. 2-24, andV= 12 above it. Parabolas adjusted to the tip radius are not a good fit to the profiles.
Figure 2-26.Time sequence of a dendritic array at
V= 40,
e= 0.1, corresponding to point (c) in Fig. 2-
24. The starting structure corresponds toV= 12, sim-
ilar to Fig. 2-25. The cellular array quickly converges
to a stationary side-branching mode of operation.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

185. 2.5 Directional Solidification 133
free dendrites at relatively low speeds of
V=12 shown in Fig. 2-25. In this case, the
side branches are just beginning to show
up, while the tip is not very noticeably par-
abolic.
Up to this point these investigations
were done at a constant anisotropy of
e= 0.1 of the capillarity length. In Fig.
2-28 the normalized supercooling
Dhas
been plotted against velocityV, where
Figure 2-27.RatiosR
tip/R
xof tip radius computed numerically (Fig. 2-24) over two theoretical predictions,
whereR
xis either the Ivantsov radius (circles) or the radius from solvability theory (asterisks). See also Fig.
2-12. The result is in nearly perfect agreement with the solvability theory down to very low speeds in the cellular
region. The Ivantsov radius (for free growth) is not a good approximation there, as the diffusion fields of neigh-
boring cells strongly overlap. At high speeds, essentially free dendritic growth is confirmed (Saito et al., 1990).
Figure 2-28.Supercooling at the tip of a cell or dendrite vs. pulling speed as obtained from numerical simula-
tion. For a flat interface at low speeds, the global conservation law forces
D= 1, then it decreases untill≈ l, and
finally approaches the slowly increasing relation
D(V) obtained for the free dendritic case (see also Fig. 2-40).
The expected dependence on capillary anisotropy
eis also recovered.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

186. D= 1 for a flat interface at stationary
growth. Two sets of data for
e= 0.1 and
e= 0.2 are shown. If we increase the pull-
ing speed above the critical valueV≈1, the
supercooling at the tip of the cellular pat-
terns first decreases, because the forward
bulges come into a range of higher tem-
perature. At intermediate velocity,
Dgoes
through a minimum and finally approaches
the broken lines corresponding to the scal-
ing relation, Eq. (2-68), together with the
Ivantsov relation, Eq. (2-56). At intermedi-
ate velocities, the supercooling
Dis above
the corresponding curve meaning that the
Péclet number, and therefore, the Ivantsov
radius is larger than expected from the free
dendritic scaling.
This is in agreement with Fig. 2-27
shown above. The minimum of the
Dver-
susVrelation is in the range where the dif-
fusion length is comparable to the cell
spacing, as expected from Fig. 2-24.
As a final example, Fig. 2-29 shows a
dendritic array at the relatively high veloc-
ityV= 40 at anisotropy
e= 0.2. As in free
dendritic growth, the structure appears
sharper than the structure in Fig. 2-26 at
smaller anisotropy.
The opposite case of extremely small an-
isotropies has not yet been analyzed in
great detail, and it is rather unclear what
happens both from a theoretical and an ex-
perimental point of view. It is likely that at
zero anisotropy
e= 0, the cells will tend to
split if the cell spacing becomes much
larger than the diffusion length, which af-
fords the possibility for chaotic dynamics
at high speeds. However, this is still specu-
lative.
Let us take a quick look at the kinetic co-
efficient
bin Eq. (2-111). As can be con-
cluded from its multiplication byV
^,bbe-
comes more and more important at high
growth rates. For the free dendritic case
with kinetic coefficient
band 4-fold an-
isotropy
b
4of the kinetic coefficient, a
scaling relation similar to Eq. (2-68) was
derived by Brener and Melnikov (1991):
(2-123)
with a constant prefactor
s
b≈5 and with
Péclet numberPas used before in Eq. (2-
68). The scaling relation, Eq. (2-123), con-
sists of several non-trivial power laws;
only the one withP=R/lrelating tip radius
to velocity has been confirmed (Classen
et al., 1991). With regard to the general
agreement between analytical and numeri-
cal results obtained so far, however, there
is little doubt that these scaling results (and
others given by Brener and Melnikov,
1991) will also hold for the dendritic re-
gion
l∂lin directional solidification.
bs
b
b
bV
D
d
P=
L2
0
92
11 2
4
72
⎛
⎝
⎜
⎞
⎠
⎟
/
//
134 2 Solidification
Figure 2-29.Pronounced parabolic directional den-
drites at speedV=40,
e= 0.2, as used in Fig. 2-28.
Note that the tip-radius here is about 0.03 and the
short-wavelength limit of neutral stability 0.05 in
units of the cellular spacing. This is qualitatively
consistent with experimental observations of large
interdendritic spacings in units of tip radii. Tip split-
ting was only observed at much lower values of cap-
illary anisotropy.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

187. 2.5 Directional Solidification 135
A final point to be kept in mind is that
the tip supercooling
Din directional solid-
ification is not small, as required by the ap-
proximations used for the derivation of the
scaling relation. On the other hand,P=1
corresponds to a supercooling as large as
D≈0.75; 0.6 for 2-dim and 3-dim, respec-
tively, and the scaling relations can be ex-
pected to hold over a large range of veloc-
ities, as already indicated from the other
free dendritic case, Fig. 2-12.
In summary, these investigations have
shown that there appears to be a smooth
transition from cellular to dendritic struc-
tures. The dendritic growth laws are very
well represented by the scaling relations
for the free dendritic case. This scaling
should hold in the region
d
0<R≈l< l<l
T (2-124)
where
lis the primary cell spacing. It was
proposed by Karma and Pelce (1989a, b)
that the transition from cells to dendrites
could occur via an oscillatory instability,
for which the present investigations under
quasi-stationary approximation have shown
no evidence. A fully time-dependent calcu-
lation is possible in principle with Green’s
function methods (Strain, 1989).
2.5.5 The Selection Problem
of Primary Cell Spacing
An important question from an engineer-
ing point of view appears to be the follow-
ing: suppose we know all the material pa-
rameters and the experimentally controlla-
ble parameters like thermal gradient and
pulling speed for a directional solidifica-
tion process – can we then predict the dis-
tance between the cells and dendrites?
A positive answer to this question is de-
sirable because the mechanical properties
of the resulting alloy are improved with a
decrease in the primary cell spacing (see
Kurz and Fisher (1998) and references
therein).
In a rigorous sense the answer is still
negative, but at least arguments can be
given for the existence of some boundaries
on the wavelengths (or cell spacings)
which can be estimated with the use of sim-
plifications.
The situation here shows some similarity
to the formation of hydrodynamic periodic
roll patterns (Newell and Whitehead, 1969;
Kramer et al., 1982; Riecke and Paap,
1986). In a laterally infinite system, a
whole band of parallel rolls is present
above the threshold for roll formation, the
so-called Eckhaus band. This was men-
tioned in Sec. 2.5.3 for directional solidifi-
cation.
The reason for the stability of these rolls
is that an infinitesimal perturbation is not
sufficient to create or annihilate a roll, but
a perturbation must exceed a threshold
value before such an adjustment can occur.
In directional solidification, the situation
is different insofar as the envelope over the
tips of the cells could make a smooth defor-
mation of very long wavelength, thereby
building up enough deformation energy so
that a cell could be created or annihilated at
isolated points. One indication for such a
process is the oscillatory instability of cells
postulated by Karma and Pelce (1989) and
Rappel and Brener (1992).
The only hard argument for the selection
of a unique wavelength comes from an
analysis of a spatially modulated thermal
gradient acting on a cellular pattern of
small amplitude (normal bifurcation)
which imposes arampon the pattern
(Misbah, 1989; Misbah et al., 1990). The
idea originally proposed for the hydrody-
namic case (Kramer et al., 1982) is to have
a periodically varying thermal gradient
parallel to the interface, which keeps the
interface flat in some regions and allowswww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

188. for the formation of cells in between
(Fig. 2-30). For such a specific setup it was
shown (Misbah et al., 1990) that a unique
wavelength must be selected in the center
of the small-gradient area. The reason for
this special construction is that it allows for
the formation of cells at arbitrarily small
amplitudes (and therefore small pinning
forces) in the region of strong thermal gra-
dient.
In general, however, the boundary con-
ditions on the other sides of the cells, due
to the container walls, are not well speci-
fied and typically will not provide such a
ramp structure (see Misbah (1989), how-
ever, for growth in a rotating vessel). For
the time being, we can therefore try to at
least find some boundary similar to the
Eckhaus band for the limits of large and
small wavelengths in the cell spacing.
It is not easy to extend the correspond-
ing analysis of small-amplitude cells
(Brattkus and Misbah, 1990) to cells with
deep grooves, as these essentially infinite
grooves present a kind of topological con-
straint on the number of longitudinal cells
in a given lateral interval. The creation or
annihilation of cells is therefore likely to
be a discrete process.
A natural mechanism for the local reduc-
tion of cell spacings (or creation of a new
cell) is either a nucleation in one of the
grooves (the liquid is supercooled), or even
more likely, the formation of a new cell out
of a side branch in such a groove. Alterna-
tively, tip splitting of a cell may give the
same result (Fisher and Kurz, 1978, 1980).
The opposite mechanism for the increase
of cell spacing (or annihilation of an exist-
ing cell) could occur through the competi-
tion of neighboring cells for the diffusion
field, such that one cell finally moves at a
slightly slower speed than the neighboring
cells and, consequently, will be supressed
relative to the position of the moving front.
These two mechanisms have been con-
jectured by many authors in the past. Some
progress has been made recently by the
confirmation of the scaling relations in
the dendritic region. It seems, therefore,
worthwhile to reformulate those conjec-
tures with the help of these scaling rela-
tions. Let us first consider the short-wave-
length
l(cell spacing) argument. Assume
that we are at dendritic growth speeds, Eq.
(2-124), ignoring here kinetic coefficients.
The solidification front then looks like an
array of individual dendrites which only
weakly interact with each other through the
diffusion fieldl<
l.
The solidification front
z= 0 will be
understood here as a smooth envelope
touching all the dendrite tips, so that defor-
mations of the front have a smallest wave-
length
lequal to the cell spacing. There are
now basically two “forces” acting on de-
formations∂
z(x,t)/∂tof that front. If some
of the tips are trailing a little behind the
others, they will be screened through the
diffusion field of the neighboring tips,
as in the conventional Mullins–Sekerka
instability, but now without a stabilizing
136 2 Solidification
Figure 2-30.Numerical study of cellular wave-
length selection at the interface by introducing a
ramp in the thermal gradient field. A high thermal
gradient on the side approximately normal to the
interface keeps the interface flat, the smaller gradient
in the center allows for cells to develop. At fixed
ramp profile a unique cell spacing is selected in the
center, starting from different initial conditions (Mis-
bah, 1990, unpublished).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

189. 2.5 Directional Solidification 137
surface tension interacting between neigh-
boring tips. Taking this into account, the
destabilizing force isF
d≈W
˜
k
(d)z
˜
kwith
W
˜
k
(d)=V|k|for a sinusoidal perturbation of
amplitude
z
˜
kof a plane interface without
surface tension moving at velocityVand
wavenumberk. The maximum lies at
k=2p/
l. The actual area under this pertur-
bation
z
˜
kcontained in the solidcellsis
smaller by a factor≈2R/
l. We thus arrive
at a maximal destabilizing force of
F
d≈W
(d)
z
l;W
(d)
≈4pVR/ l
2
(2-125)
corresponding to a depression or enhance-
ment of every second dendrite.
On the other hand, each of these individ-
ual dendrites knows its operating point,
and through the given velocity its super-
cooling at the tip. We approximate this
by the asymptotic form of Eq. (2-56)
D≈1–1/Psince basically the variation
of
DwithPenters below, even throughP
may not be very large compared to unity.
Capillary effects do not seem to be very
important in this region and are thus ne-
glected here for simplificity. By definition,
D=1–z
tip/l
T, and the two expressions for
Dcan be evaluated:z
tip≈l
Tl/R. The stabi-
lizing forceF
s≈W
(s)
z
lfollows from the
obvious relation
W
(s)
=dV
tip/dz
tipas
(2-126)
As expected, the sign of this “force” is op-
posite to the destabilizing force, Eq. (2-
125). Setting the sum of the two equal to
zero, we expect an instability to occur first
at cell spacings
(2-127)
for segregation coefficients around one.
We cannot say much about small segrega-
tion coefficients because the nonlinearity
in Eq. (2-114) replacing Eq. (2-56) be-
l⎡ll
T
F
VR
ll
T
s
ss
≈≈ −WW
() ()
;z
l 2
comes important there. Of course, a num- ber of rough approximations were used specifically in the treatment of the destabi- lizing force, but this argument should at least qualitatively capture the competition mechanism between neighboring den- drites. A more detailed analysis (Warren and Langer, 1990) is quite promising at large velocities in comparison with experi- ments (Somboonsuk et al., 1984) (see also Kessler and Levine, 1986c; Bechhoefer and Libchaber, 1987).
Let us now look at the large-wavelength
limit
l. The initial growth conditions are
assumed to be just as before, Eq. (2-124), but now at possibly large cell spacings
l.In
the numerical calculations, it was found (Saito et al., 1989) that for fixed cell spac- ing
lat increasing velocity, atail instabil-
ityoccurs (Fig. 2-31). A side branch in the
Figure 2-31.After a sudden increase in the growth
rate in the dendritic region at fixed cell spacing, tail
instability occurs. One of the side branches near the
tip produces a proturberance in the forward direction,
which then becomes a new primary cell (the imposed
mirror symmetry is not present in reality of course).
In accordance with the stability of dendritic cells
against tip splitting (compare Fig. 2-29), this tail in-
stability appears to be an important selection mecha-
nism for primary cell spacing. See also Fig. 2-43.www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

190. groove between two dendrites splits off a
ternary branch, which then moves so fast in
the strong supersaturation in the groove
that it finally becomes a new primary
branch. In the plot, the imposed mirror
symmetry is, of course, artificial, but it
even acts opposite to this effect, making
the process more plausible in reality. In
fact, this is also observed experimentally
(Esaka and Kurz, 1985), in particular when
the solid consists of slightly misaligned re-
gions separated by grain boundaries, so
that the growth direction of two neighbor-
ing dendrites is slightly divergent. Our ba-
sic assumption now is that this tail instabil-
ity occurs when the intersection of para-
bolic envelopes over neighboring dendrites
occurs at a pointz⎡0, where, theoretically,
D>1. In this case, there is no need for
long-range diffusion around a side branch,
for its dynamics become local. Of course,
this assumption ignores geometrical com-
petition between neighboring side branches
to a certain extent, but for the moment
there seems to be no better argument to
hand.
Taking into account the point that neigh-
boring parabolas with tip position atz
tip>0
cannot intersect further down to the cold
side than atz= 0, we obtainz
tip=l
Tl/R
from the two relations for
D, just as in the
previous case of small wavelengths. But
now we must use the parabolic relation
z
tip=l
2
/8Rfor the intersection of two pa-
rabolas of radiusRatz= 0, which are a dis-
tance
lapart. The tail instability is then ex-
pected to occur for
(2-128)
again with a prefactor roughly of the order
of unity. In comparison with Eq. (2-127), it
can be seen that in both cases the same
scaling relation results. The scaling with
the inverse growth ratelfollows the neu-
tral stability curve at velocities safely in
l∂ll
T
between the two critical values and again recovers Eq. (2-1) by noting thatl~V
–1
and thereforeV l
2
≈constant. The results,
Eqs. (2-127) and (2-128), seem to be in agreement with experiments (Somboonsuk et al., 1984; Kurowsky, 1990; Kurz and Fisher, 1981, 1998) concerning the scaling with respect to diffusion lengthland ther-
mal lengthl
T. The limitkÆ0 for the seg-
regation coefficient as a singular point is not reliably tractable here.
The previously given relation
l~l
1/4
(Hunt, 1979; Trivedi, 1980; Kurz and Fisher, 1981) seems to be valid in an inter- mediate velocity region (Fig. 2-28) where
Ddoes not vary significantly, so that
z
tip/l
T≈1/2 (see also Sec. 2.6.1).
A serious point is the neglect of surface
tension and anisotropy in these derivations. In the experiments analyzed so far the rela- tionV
l
2
≈const. seems to hold approxi-
mately, but what happens for the capillary anisotropy
egoing to zero? Numerically,
tip splitting occurs at lower velocities for smaller
e. In a system with anistropye=0
(and zero kinetic coefficient) the structures probably show chaotic dynamics at veloc- ities where the diffusion lengthlis smaller
than the short wavelength limit of the neu- tral stability curve (Fig. 2-24), but this is rather speculative (Kessler and Levine, 1986c).
In considering whether the tail instabil-
ity (large
l) or the competition mechanism
(small
l) will dominate in casting pro-
cesses, we tend to favor the former. If the solidification front consists of groups of dendrites slightly misoriented against each other due to small-angle grain boundaries, cells will disappear at points where the lo- cal growth directions are converging and new cells will appear through the tail in- stability at diverging points at the front.
To summarize, the most likely scaling
behavior of the primary cell spacing
l,de-
138 2 Solidificationwww.iran-mavad.com
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191. 2.5 Directional Solidification 139
pending on pulling velocities, follows Eqs.
(2-1) and (2-128) as a consequence of the
arguments presented in this section. This
conclusion is supported by a number of ex-
periments (Billia et al., 1987; Somboonsuk
et al., 1984; Kurowsky, 1990; Esaka and
Kurz, 1985), but more work remains to be
done.
2.5.6 Experimental Results
on Directional Dendritic Growth
Since 1950,in situexperiments on direc-
tional solidification (DS) of transparent
model systems have been performed (Kof-
ler, 1950). However, it was some time be-
fore such experiments were specifically
conceived to support microstructural mod-
els developed in the 1950s and early 1960s.
The work of Jackson and Hunt (1966) is a
milestone in this respect (see also Hunt et
al., 1966). Their experimental approach to
dendritic growth has been developed fur-
ther by several groups: Esaka and Kurz
(1985), Trivedi (1984), Somboonsuk et al.
(1984), Somboonsuk and Trivedi (1985),
Eshelman et al. (1988), Seetharaman and
Trivedi (1988), Seetharaman et al. (1988),
de Cheveigne et al. (1986), Akamatsu et al.
(1995), Akamatsu and Faivre (1998), and
others. Substantial progress has been made
during these years, especially due to results
obtained by Faivre and coworkers in very
thin (~ 15 µm) cells that constrain the pat-
terns to two dimensions. This research is
still producing interesting new insights into
the dynamics of interface propagating dur-
ing crystallization.
The specific interest of DS is that growth
morphologies can be studied not only for
dendritic growth but also for cellular and
plane front growth. We will discuss these
phenomena in the sequence of their appear-
ance when the growth rate is increased
fromV
c, the limit of first formation of
Mullins–Sekerka (MS) instabilities (also
called limit of constitutional undercool-
ing), to rates where plane front growth
again appears at velocities aboveV
a, the
absolute stability limit.
Morphological instabilities. The onset
of plane front interface instability is ob-
served to start at defects such as grain
boundaries, subgrain boundaries and dislo-
cations, forming a more or less pronounced
depression at the intersection with the
solid–liquid boundary, as shown in Fig.
2-32 (Fisher and Kurz, 1978). It is inher-
ently difficult to quantitatively observe the
break-down because the growth rates are
small, and a long period of time is required
to reach steady state.
The amplitude and wavelength of the
perturbations as a function ofVdeveloping
in systems with small distribution coeffi-
cientsk(of the order of 0.1) are shown in
Fig. 2-33. In CBr
4–Br
2(de Cheveigne et
al., 1986) and SCN–ACE (Eshelman and
Trivedi, 1987), the bifurcation is of a sub-
critical type, i.e., there are two critical
growth rates, one for increasing growth
rate,V
c
+, and another lower value for de-
creasing growth rate,V
c
–. Therefore, a peri-
odic interface shape with infinitesimally
small amplitudes cannot form in these
systems. As has been discussed in Sec.
2.5.3, only systems withknear unity will
give supercritical (normal) bifurcation
with a single, well-defined critical growth
rateV
c. The evolution of the wavelength
for two temperature gradients (70 and
120 K/cm) is shown in Fig. 2-33b. At the
onset of instability the experimentally de-
termined wavelength is smaller than the
critical wavelength by a factor of 2–3 com-
pared with linear stability analysis. In-
creasing the rate above the threshold leads
to a decrease in the wavelength proportional
toV
–0.4
(de Cheveigne et al., 1986; Kurow-
sky, 1990). Once stability has started, thewww.iran-mavad.com
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192. structure evolves to a steady-state cellular or
dendritic growth mode. Which one of these
structures will finally prevail is a question of
the growth conditions.
Cells and dendrites. Losert et al. (1998a)
developed an interesting technique in
which a spatially periodic UV laser pulse is
directed onto the solidification front of the
140 2 Solidification
Figure 2-32.Morphological instabilities of a planar solid/liquid interface as seen on an inclined solidification
front between two glass plates (arrows). Photograph (a) was taken at an earlier stage than (b). The beginning of
the breakdown at defects such as dislocations, subgrain boundaries, or grain boundaries intersecting with the
solid/liquid interface is evident. The widths of the photographs correspond to 100 µm.
(a)
Figure 2-33.Amplitude,A, and wavelength, l, of a periodic deformation of the solid/liquid interface of
CBr
4–Br
2solution versus pulling speed (de Cheveigne et al., 1986). Diagram (a) shows the hysteresis between
appearance and disappearance of perturbations (typical for an inverse bifurcation) for a temperature gradient of 12 K/mm. The lines in diagram (b) represent the calculated neutral stability curves for the two gradients indi- cated. Open circles are experimental results for 7 K/mm, and crosses, for 12 K/mm. See also Fig. 2-21.
(b)
(a) (b)
J
J
j
jwww.iran-mavad.com
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193. 2.5 Directional Solidification 141
transparent succinonitrile–coumarin sys-
tem. These experiments allowed for a
systematic investigation of the dynamic se-
lection and stability of cellular structures.
Through an increase inV(orC
•,orade-
crease inG
T), a columnar dendritic struc-
ture can be formed out of a cellular array
(Fig. 2-34). All three morphologies (in-
stabilities, cells, dendrites) appear to have
their own wavelength or array spacing.
Owing to competition between neighboring
crystals, the meanspacing of large ampli-
tude cells seems to be always larger than
that of the initial perturbations of the plane
front, and the mean trunk distance (primary
spacing
l) of the dendrites, larger than the
Figure 2-34.Time evolution of the solid/liquid interface morphology when accelerating the growth rate from
0 to 3.4 µm/s at a temperature gradient of 6.7 K/mm. Magnification 41¥. (a) 50 s, (b) 55 s, (c) 65 s, (d) 80 s,
(e) 135 s, (f) 740 s (Trivedi and Somboonsuk, 1984).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

194. spacing between smooth cells. The reason
for this change in typical spacing is not yet
clear. We will come back to this subject
later. Before we do so, some relevant obser-
vations on tip growth need to be discussed.
The tip is the “head” of the dendrite
where most of the structural features are in-
itiated. Fig. 2-35 shows a dendrite tip of
succinonitrile (SCN) with 1.3 wt.% ace-
tone (ACE) in an imposed temperature gra-
dient,G
T= 16 K/cm and a growth rate,
V= 8.3 µm/s. The smooth tip of initially
parabolic shape (Fig. 2-35b) is clearly vis-
ible. In contrast of free thermal dendrites
(Fig. 2-9), in DS of alloys the secondary
instabilities start forming much closer to
the tip. The imposed temperature gradient
also widens the dendrite along the shaft
relative to a parabola fitted to the tip. This
effect increases with an increasing temper-
ature gradient (Esaka, 1986).
142 2 Solidification
Figure 2-35.Dendrite tip of
SCN–1.3 wt.% acetone so-
lution in directional growth.
(a) ForV= 8.3 µm/s and
G= 1.6 K/mm; (b) parabola
fitting the tip growing
atV= 33 µm/s andG=
4.4 K/mm (Esaka and Kurz,
1985; Esaka, 1986).
(b)
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195. 2.5 Directional Solidification 143
The sequence of steady-state growth
morphologies, from well-developed large
amplitude cells to well-developed den-
drites, is shown in Fig. 2-36. Besides the
information on the form and size of the cor-
responding growth morphologies, this fig-
ure also contains indications specifying the
diffusion lengthl=2D/Vand the ratio of
the half spacing over tip radius. The char-
acteristic diffusion distance decreases more
rapidly than the primary spacing of the den-
drites (Fig. 2-36). WhenlÛ
lthe ratiol/2R
of directionally solidified SCN–1 wt.% ACE
alloys is between 5.5 and 6, in agreement
with numerical calculations (Sec. 2.5.3).
These observations are summarized in
Fig. 2-37. Three areas of growth can be dif-
ferentiated for this alloy:
i) at low speeds, cells are found showing
no side branches and a non-parabolic tip;
ii) at intermediate rates (over a factor 5 in
V), cellular dendrites are formed with weakly
developed secondary arms, and they show
an increasingly sharpened parabolic tip;
iii) at large rates dendritic arrays grow
with well-developed side branches and a
tip size much smaller than the spacing.
It is difficult at present to judge the influ-
ence of the width of the gap of the experi-
Figure 2-36.Cellular and den-
dritic growth morphologies in
SCN–1.3 wt.% acetone; thermal
gradient 8 K/mm <G< 10.5 K/mm
(Esaka, 1986). The growth rateV
in µm/s, the diffusion length
(2D/V) in mm, and the ratio of the
primary trunk spacing to the tip
diameter are as follows: A = 1.6,
1.6, 2.0; B = 2.5, 1.0, 2.5; C = 8.3,
0.3, 5.5; D = 16, 0.16, 6.0; E = 33,
0.08, 7.5; F = 83, 0.03, 9.0.www.iran-mavad.com
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196. mental cell, which had approximately the
same size as the diffusion distance, when
l=l. The corresponding growth rate also
marks the transition from two- to three-di-
mensional growth of dendrites, as can be
seen in Fig. 2-36. Under conditions A to C,
no secondary arms are observed perpendic-
ular to the plane of observation, while con-
ditions D to F show well-developed 3D
dendrites even far behind the tip. There-
fore, the gap might somewhat influence the
values of the transition rates but not the
qualitative behavior of the transition. As
mentioned above, the theory (in quasi-sta-
tionary approximation) only indicates a
very gradual change in morphology. It is
interesting to compare Fig. 2-36 with Figs.
2-25 and 2-26. It can be seen that both the
theoretical and the experimental ap-
proaches show qualitatively the same be-
havior, even if the material constants used
are not the same.
The critical role of crystalline anisotropy
in interface dynamics has been demon-
strated experimentally in directional growth
of transparent CBr
4–C
2Cl
6alloys grown as
single crystals in thin samples (Akamatsu
et al., 1995). With the·100Òdirection
oriented along the heat flow direction,
characteristic cellular/dendritic arrays such
as shown in Fig. 2-36 are obtained. With
other orientations, where growth is ren-
dered “effectively isotropic” a “seaweed”
structure is observed, the tips of which
form what has been called on the basis of
theoretical predictions a doublon (Ihle and
Müller-Krumbhaar, 1994).
The formation of doublons has also been
suggested to play an important role in the
formation of “feathery” grains in technical
aluminum alloys. Recent detailed observa-
tions with electron back-scattered diffrac-
tion (EBSD) combined with optical and
scanning electron microscopy (Henry et
al., 1998) have clearly shown that feathery
grains are made of·110Òcolumnar den-
drites, whose primary trunks are aligned
along and split in their center by a (111)
coherent twin plane. The impingement of
secondary·110Òside arms gives rise to in-
coherent wavy twin boundaries. The switch
from·100Òto·110Ògrowth direction was
attributed to the small anisotropy of the
solid–liquid interfacial energy of alumi-
144 2 Solidification
Figure 2-37.Primary trunk spacing, diffusion
length and tip radius of SCN–1.3 wt.% ace-
tone dendrites as a function of the growth rate
for a temperature gradient of 9.7 K/mm. The
various points correspond to the conditions
and microstructures given in Fig. 2-36 (Esaka,
1986). In region A no side arms are observed
and the tips are of non-parabolic shape; in B
the tip becomes parabolic and some side arms
appear; in C well-developed side arms and a
parabolic tip are the sign of isolated tips.www.iran-mavad.com
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197. 2.5 Directional Solidification 145
num, which can be changed by the addition
of solute (Henry et al., 1998).
The scaling of the initial side-branch
spacing
l
2with respect to the tip radius is
shown in Fig. 2-38. Both quantities scale
closely withL
2
V= const. orL
2
C
•= const.
(whereL~Ror
l
2). Fig. 2-38b also shows
the quantity
l
p, the distance from the tip
down the shaft, where the first signs of tip
perturbations in the SCN–ACE system can
be observed. Here
l
pis of the same order
as
l
2.
The ratio
l
2/Robtained by Esaka and
Kurz (1985) for SCN–1.3 wt.% ACE is in-
dependent of growth rate according to the
precision of the measurements and takes a
value of 2.1 ± 0.2 (see Fig. 2-39 and Table
2-1). This is in good agreement with the
measurement of Somboonsuk and Trivedi
(1985) who found a value of 2.0 on the
same system over a wide range of growth
rates, compositions, and temperature gra-
dients (Trivedi and Somboonsuk, 1984).
This ratio increases with crystal anisotropy
(a)
(b)
Figure 2-38.Characteristic dimensions
(tip radius,R, initial secondary arm spac-
ing,
l
2, and distance of appearance of
secondary instabilities,
l
p) as a function
of growth rate (a), and as a function of
the alloy concentration (b) (Esaka, 1986).www.iran-mavad.com
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198. in agreement with numerical calculations
(Figs. 2-26, 2-29, 2-13) and decreases with
increasing temperature gradient. On the
other hand, its value is larger in the case of
free dendritic growth (Table 2-1).
The tip undercooling of the dendrite is a
measure of the driving force necessary for
its growth at the imposed rateV. Fig. 2-40
shows the variation of the tip temperature
withV, the undercooling being defined by
the difference betweenT
mand the tip tem-
perature. During an increase in growth rate,
the cellular growth region is characterized
by a decrease in undercooling, while to-
ward the dendritic region, the undercooling
increases again (Fig. 2-28).
One of the characteristics of directional
solidification of cells/dendrites is the for-
mation of array structures with a primary
trunk spacing
l. For cellular and cellu-
lar–dendritic structures the growth direc-
tion/crystal orientation relationship plays
an important role in the establishment of
stable or unstable array patterns with a
146 2 Solidification
Figure 2-39.Ratio of initial arm spac-
ing to tip radius as a function of growth
rate (Esaka and Kurz, 1985).
Figure 2-40.Tip temperatures of
the dendrites of Fig. 2-36 as a function of growth rate (Esaka and Kurz, 1985). See also Fig. 2-28.www.iran-mavad.com
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199. 2.5 Directional Solidification 147
characteristic spacing, as has been shown
by Akamatsu and Faivre (1998). The
growth-rate dependence of dendritic pat-
terns has been shown in the log–log plot of
Fig. 2-37. We can see that the slope of
lvs.
Vis initially lower than the slope ofRvs.
V. If we take the three points around
V(l=
l), where the tip temperature is ap-
proximately constant, the slope is about
0.25, while a mean slope through all mea-
surements shown gives 0.4. This is fully
consistent with data by Kurowsky (1990).
Furthermore, it is in agreement with our ar-
guments in Sec. 2.5.5.
Most of the measurements give a rate ex-
ponent somewhere between the two limit-
ing values, 0.25–0.5. Taking the lower
value expressed by temperature gradient
and concentration gives a relationship
l~G
T
–0.5C
•
0.25(Hunt, 1979; Kurz and
Fisher, 1981; Trivedi, 1984). Therefore, a
normalized spacing
l
4
G
2
T
V/kDT
0is plot-
ted vs.Vin Fig. 2-41, showing that differ-
ent materials behave similarly except for a
constant factor (note that the solidus–liq-
uidus interval of an alloy,DT
0, is propor-
tional toC
•).
Primary spacings, however, are not
uniquely defined but form a rather wide
distribution. This is shown in Fig. 2-42 for
one superalloy which was directionally so-
lidified under different conditions. This be-
havior can be understood by examining the
mechanism of wavelength (spacing) reduc-
tion through tail instability, which is a
complicated process. A series of competi-
tive processes between secondary and
tertiary arms in a region behind the tip (of
the order of one primary spacing) finally
causes one tertiary branch to grow through
and to become a new primary trunk (Esaka
et al., 1988), as shown in Figs. 2-43 and
2-31. The range of stability of dendritic
arrays has been analysed by Losert et al.
(1998b). Hunt and Lu (1996) approached
Figure 2-41.Normalized primary trunk spacing, l,
as a function of growth rate for various alloy systems
(Somboonsuk et al., 1984). Note that the variation
with growth rate at these high velocities is in agree-
ment with the discussion in Sec. 2.5.5.
Figure 2-42.Distribution of nearest dendrite–den-
drite separations (primary trunk spacings) measured on a transverse section of a directionally solidified Ni-base superalloy (Quested and McLean, 1984). Solid lines for 16.7 µm/s and broken lines for 83.3 µm/s.www.iran-mavad.com
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200. this problem by numerical modeling using
a minimum undercooling criterion for
the selection of the lower limit of array
growth.
Effects at high growth rates
As the final topic in this section, we dis-
cuss some interesting effects which have
been observed at very high growth rates.
In laser experiments of the type shown in
Fig. 2-7, the interface may be driven to ve-
locities of several m/s. Under such high
rates, the structures become extremely
fine. Primary spacings as small as 20 nm
have been measured in Al–Fe alloys (Gre-
maud et al., 1990). Fig. 2-44 represents
measured primary spacings (black squares),
measured secondary spacings (open
squares), and the calculated tip radius of
the dendrites, using Ivantsov’s solution
and the solvability-scaling criterion (Kurz
et al., 1986, 1988).
landl
2vary asV
–0.5
,
as does the tip radius when the Péclet num-
ber of the tip is not too large. At large
Péclet numbers (or small diffusion dis-
tances), capillary forces become dominant,
which is the reason for the limit of absolute
stability,V
a. Ludwig and Kurz (1996a) de-
termined cell spacing and amplitude close
to absolute stability in succinonitrile–
argon alloys and found for both
l
3/2
V=
const.
Since the classical paper by Mullins and
Sekerka (1964), it has been known that
plane front growth should also be observed
at interface rates where the diffusion length
reaches the same order as the capillary
length. This critical rate, called the limit of
absolute stability, can be calculated from
linear stability analysis (for temperature
148 2 Solidification
Figure 2-43.Mechanism for the formation of a new
primary trunk by repeated branching of side arms at a
grain boundary (gb). The increasing spacing at the gb
allows a ternary branch to develop and to compete in
its growth with other secondary branches (hatched
arms) (Esaka et al., 1988).
Figure 2-44.Experimentally determined
primary spacings (black squares), secon- dary arm spacings (open squares) and calculated tip radius (line) for Al–4 wt.% Fe alloy rapidly solidified by laser treat- ment. The minimum in tip temperature or the maximum in undercooling is due to the decreasing curvature undercooling when the dendrite approaches the limit of absolute stability,V
a(Gremaud et al.,
1990).www.iran-mavad.com
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201. 2.5 Directional Solidification 149
gradients which are not too high) approxi-
mately as
V
a=DT
0D/kG
HereGis the ratio of solid–liquid interface
energy to specific melting entropy. Typical
limits are of the order of m/s. The precise
value depends also on the effect of a vary-
ing solute diffusion coefficient (due to the
large undercoolings), on the variations ofk
andDT
0with the growth rate associated
with the loss of local equilibrium, and, fi-
nally, on interface kinetics, which cannot
be ignored. According to an extensive
study of several alloy systems, it may be
stated that atV
awe generally do not ob-
serve a simple plane front growth but
rather an oscillating interface, which pro-
duces bands of plane front and cellular
dendritic morphology. Fig. 2-45a shows
such a transition from columnar dendritic
grains to bands (Carrard et al., 1992).
Fig. 2-45b shows more details of the
banded structure. It is visible that the dark
band is cellular/dendritic and the clear
band structure-less (supersaturated plane
front growth). The possibility of chaotic
interface motion was shown by Misbah et
al. (1991), and oscillatory motion of a
plane interface with time was suggested by
Corriell and Sekerka (1983) and Temkin
(1990). The full theoretical analysis of the
banding problem has been given by Karma
and Sarkissian (1993), see also Kurz and
Trivedi (1996). Growth rates much higher
thanV
aare needed in order to definitely
produce an absolutely stable plane solid–
liquid interface (Fig. 2-46).
Under steady-state growth conditions
using a transparent organic system, Ludwig
and Kurz (1996b) observed the onset of ab-
solute stability but no sign of banding. The
latter was due to the fact that the experi-
ments had to be undertaken with highly di-
luted alloys. This was necessary as one had
to work under heat flow limited conditions
allowing a growth rate of not more than
some mm/s. In order to reach absolute
stability at these low rates the composition
(andDT
0) had to be kept very small (typi-
cally below 100 mK).
2.5.7 Extensions
In this section, we summarize a few im-
provements on the theory of solidification
regarding various effects that are important
in the practical experimental situation.
They are observed when experimental pa-
rameters are outside the range of the simple
models considered in this chapter, or when
Figure 2-45.Transmission electron micrographs of
the banded structure in laser-remelted alloys. (a)
General view of the abrupt transition from columnar
eutectic grains (lower part) to a banded structure
(upper part) in a eutectic Al–33 wt.% Cu alloy with
a solidification rate,V= 0.5 m/s. (b) Enlarged view
of the dark and light bands in a cellular dendritic
Al–4 wt.% Fe alloy withV= 0.7 m/s (Carrard et al.,
1992).www.iran-mavad.com
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202. additional factors influence the growth of
the solid.
The models in this chapter are minimal
in the sense that they were intended to cap-
ture the essence of a phenomenon with the
smallest possible number of experimen-
tal parameters. Despite this simplification,
however, the results appear to be of general
importance.
The discussion of free dendritic growth
has been restricted so far to small Péclet
numbers or low supercoolings. In direc-
tional solidification, however, we are usu-
ally at moderate or even high Péclet num-
bers (Ben Amar, 1990; Brener and Melni-
kov, 1990). It was shown by Brener and
Melnikov (1990) that in this case a devia-
tion from dendritic scaling occurs ifP
e
1/2
increases approximately beyond unity.
At high growth rates, furthermore, kinetic
coefficients can no longer be ignored at
the interface. A proposed scaling relation
(Brener and Melnikov, 1991) was con-
firmed numerically (Classen et al., 1991),
and combined anisotropy of surface ten-
sion and kinetic coefficient was treated an-
alytically (Brener and Levine, 1991; Ben-
Jacob and Garik, 1990). A general treat-
ment of kinetic coefficients on interface
stability was given by Caroli et al. (1988).
For eutectic growth, this was formulated by
Geilikman and Temkin (1984). For higher
anisotropies than considered so far, we en-
counter facets on the growing crystals.
This was analyzed for single dendrites
(Adda Bedia and Ben Amar, 1991; Maurer
et al., 1988; Raz et al., 1989; Yokoyama
and Kuroda, 1988) and for directional so-
lidification (Bowley et al., 1989).
In our treatment of directional solidifica-
tion, only one diffusion field was treated ex-
plicitly, namely the compositional diffusion.
If a simple material grows dendritically
(thermal diffusion), small amounts of impur-
ities may become a matter of concern. This
was reconsidered by Ben Amar and Pelce
(1989), confirming the previous conclusion
(Karma and Langer, 1984; Karma and Kot-
liar, 1985; Lipton et al., 1987) that impur-
ities may increase the dendritic growth rate.
150 2 Solidification
Figure 2-46.Experimental velocities at the transi-
tions from dendrites to bands (squares) and from
bands to microsegregation-free structures (circles)
for the (a) Al–Fe, (b) Al–Cu and (c) Ag–Cu sys-
tems. The triangles indicate experimentally deter-
mined transitions from dendrites to microsegregation-
free structure, and the crosses the values of the abso-
lute stability limit. The full and dashed lines are the
theoretical predictions of the model for dilute and con-
centrated solutions, respectively (Carrard et al., 1992).www.iran-mavad.com
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203. 2.5 Directional Solidification 151
A subject of appreciable practical impor-
tance concerns the late stages of growth,
where coarsening of the side branch struc-
tures occurs together with segregation
(Kurz and Fisher, 1998). If elastic forces
are not of primary importance, it is now
generally accepted (see Chapter 6, Sec.
6.4.1) that a relationL*=A+Bt
1/3
(Huse,
1986) is a good representation of typical
length scalesL* varying with timetduring
diffusional coarsening processes. This con-
firms the classical Lifshitz–Slyozov–
Wagner theory (Lifshitz and Slyozov,
1961; Wagner, 1961). The result, however,
is not specific about the geometric details,
for example, in directional solidification,
and it also assumes constant temperature.
We have only briefly mentioned the tran-
sition from dendritic arrays back to a plane
front during directional solidification at
very high speeds. The importance of ki-
netic coefficients was demonstrated by
Brener and Temkin (1989), and recently,
the possibility of chaotic dynamics in this
region similar to those described in the Ku-
ramoto–Sivashinsky equation was sug-
gested by Misbah et al. (1991).
A richness of dynamic phenomena was
obtained in a stability analysis of eutectics
(Datye et al., 1981). The possibility of
tilted lamellar arraysin eutectics was dem-
onstrated by Caroli et al. (1990) and, simi-
larly, for directional solidification at high
speeds by Levine and Rappel (1990). This
was observed in experiments on nematic
liquid crystals (Bechhoefer et al., 1989)
and in eutectics (Faivre et al., 1989).
Finally, the density difference between
liquid and solid should have some marked
influence on the growth mode (Caroli
et al., 1984, 1989). For dendritic growth,
forced flow was treated in some detail by
Ben Amar et al. (1988), Ben Amar and Po-
meau (1992), Bouissou and Pelce (1989),
Bouissou et al. (1989, 1990) and Rabaud et
al. (1988). Other sources of convective in-
stability (Corriell et al., 1980; Sahm and
Keller, 1991) cannot be discussed here
in any detail, as the literature is too exten-
sive in order to be covered within this
chapter.
Phase-field models
In recent years, the phase-field metho-
dology has achieved considerable impor-
tance in modeling and numerically simulat-
ing a range of phase transitions and com-
plex growth structures that occur during
solidification. Phase-field models have
been applied to a wide range of materials
such as pure melts, simple binary alloys,
eutectic and peretectic phase transitions
and grain growth structures in situations as
diverse as dendritic growth and rapid solid-
ification.
In the phase-field formulation a mathe-
matically sharp solid–liquid interface is
smeared out or regularized and treated as
a boundary layer, with its own equation
of motion. The resulting formulation no
longer requires front tracking and the im-
position of boundary conditions, but must
be related to the sharp interface model by
an asymptotic analysis. In fact, there are
many ways to prescribe a smoothing and
dynamics of the sharp interface consistent
with the original sharp-interface model. So
there is no unique phase-field model, but
rather a family of related models. The first
phase-field model was developed by Lan-
ger (1986) inad hocmanner. Subsequently,
models have been placed on a more secure
basis by deriving them within the frame-
work of irreversible thermodynamics (Pen-
rose and Fife, 1990; Wang et al., 1993).
Caginalp (1989) presented for the first time
the relation between phase-field models
and sharp interface models or free boun-
dary problems by considering the sharpwww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

204. interface limit in which the interface thick-
ness is allowed to tend to zero.
However, the technique also has some
disadvantages. The first is related to the ef-
fective thickness of the diffuse interface of
alloys (1–5 nm) which is three to four or-
ders of magnitude smaller than the typical
length scale of the microstructure. Since
the interface must spread over several
points of the mesh, this limits considerably
the size of the simulation domain. The sec-
ond problem arises from the attachment ki-
netics term, which plays a significant role
in the phase-field equation, unlike micro-
structure formation of metallic alloys at
low undercooling. These two factors have
so far limited phase-field simulations of
alloy solidification with realistic solid-
state diffusivities to relatively large super-
saturations. Recent mathematical and com-
putational advances, however, are rapidly
changing this picture. Some of the recent
advances include: 1) a reformulated asymp-
totic analysis of the phase-field model
for pure melts (Karma and Rappel, 1996,
1998) that has (i) lowered the range of ac-
cessible undercooling by permitting more
efficient computations with a larger width
of the diffuse interface region (as com-
pared to the capillary length), and (ii) made
it possible to choose the model parameters
so as to make the interface kinetics vanish;
2) a method to compensate for the grid an-
isotropy (Bösch et al., 1995); 3) an adaptive
finite element method formulation that re-
fines the zone near the diffuse interface and
that has been used in conjunction with the
reformulated asymptotics to simulate 2D
dendritic growth at low undercooling (Pro-
vatas et al., 1998); 4) the implementation
of the method for fluid flow effects (Diep-
ers et al., 1997; Tonhardt and Amberg,
1998; Tong et al., 1998), and 5) the extension
of the technique to other solidification phe-
nomena including eutectic (Karma, 1994;
Elder et al., 1994; Wheeler et al., 1996) and
peritectic reactions (Lo et al., 2000).
2.6 Eutectic Growth
2.6.1 Basic Concepts
Eutectic growth is a mode of solidifica-
tion for a two-component system. Operat-
ing near a specific point in the phase dia-
gram, it shows some unique features (Kurz
and Fisher, 1998; Lesoult, 1980; Hunt and
Jackson, 1966; Elliot, 1983).
The crucial point in eutectic growth is
that the solidifying two-component liquid
at a concentration nearC
E(Fig. 2-19) can
split into two different solid phases. The
first phase consists of a high concentration
of A atoms and a low concentration of B at-
oms; the second solid phase has the oppo-
site concentrations. These two phases ap-
pear alternatively as lamellae or as fibers of
one phase in a matrix of the other phase.
One condition for the appearance of a
eutectic alloy is apparently a phase dia-
gram (as sketched in Fig. 2-19) with a tem-
peratureT
E. The two-phase regions meet at
(T
E,C
E), and the two liquidus lines inter-
sect before they continue to exist as meta-
stable liquidus lines (dash-dotted line) at
temperatures somewhat belowT
E. This is a
material property of the alloy (Pb–Sn for
example). The other condition is the ex-
perimental starting condition for the con-
centration in the high-temperature liquid.
Assume that we are moving a container
filled with liquid at concentrationC
•in a
thermal gradient field where the solid is
cold, the liquid is hot, and the solidification
front is proceeding towards the liquid in
the positivez-direction. As a stationary so-
lution, we find a similar condition to the
simple directional solidification case; that
is the concentration in the solidC
–
Saver-
152 2 Solidificationwww.iran-mavad.com
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205. 2.6 Eutectic Growth 153
aged across the front must be equal toC
•to
maintain global mass conservation together
with a stationary concentration profile near
the front.
Assume now that the liquid concentra-
tion at infinity,C
•, is close toC
Eand the
temperature at the interface is atT
E. The
solid may split into two spatially alternat-
ing phases now on the equilibrium (bino-
dal) solidus linesC
˜
S(T
I), one located near
CÆ0, and one located nearCÆ1at
T
I<T
E. To see what happens, let us look
at the situation with the concentration in
the liquidC
•being precisely at eutectic
compositionC
•=C
E.AtT
I<T
E, there may
now be alternating lamellae formed of
solid concentrationC
˜
S
aandC
˜
S
b(Fig. 2-47),
the corresponding metastable liquidus con-
centrations being atC
˜
L
aandC
˜
L
b. For both
solid phases now the liquid concentration
C
•=C
Eis in the metastable two-phase
regions atT
I:C
˜
S
a<C
•<C
˜
L
a,C
˜
L
b<C
•<C
˜
S
b.
In principle, we have to consider the pos-
sibility of metastable solid phases (Tem-
kin, 1985), which we ignore here for sim-
plicity.
Diffusion of excess material not incorpo-
rated into one of the lamellae does not have
to continue up to infinity but only to the
neighboring lamella, which has the oppo-
site composition relative toC
•. Assuming
again for simplicity a symmetric phase dia-
gram, we may write this flux balance as
(2-129)
where the left-hand side describes the flux
Jof material to be carried away from each
lamellar interface during growth at veloc-
ityV, which then must be equal to the
diffusion current (concentration gradient)
over a distance
l/2 to the neighboring la-
mella (across the liquid, as we ignore diffu-
sion in the solid).DC
˜
=C
˜
L
(a)–C
˜
L
(b)is the
concentration difference in the liquid at the
interfaces of the lamellae, andk<1 is the
segregation coefficient.
With the help of the liquidus slopes
dT/dC
˜
Las material parameters, we can ex-
press the undercoolingD
T
I=T
E–T
Ias
(2-130)
(symmetrical phase diagram assumed
aboutC
E) and thus
(2-131)
Again,l=2D/Vis the diffusion length.
So far we have ignored the singular
points on the interface where the liquid and
two lamellaeaandbmeet (Fig. 2-47). At
this triple point in real space (or three-
phase junction), the condition of mechani-
cal equilibrium requires that the surface
tension forces exerted from the three inter-
faces separatinga,band the liquid cancel
to zero. In the simplest version, this condi-
tion defines some angles
J
a,J
bof the two
DT
T
C
Ck
l
I
L
E=
d
d
21
˜
()−
l
DDTC
T
C
I
L=
d
d
˜
˜
2
VC k D
C
E =()
˜
(/)
1
2
−
D
l
Figure 2-47.Sketch of symmetrical eutectic phase
diagram (a) and eutectically growing lamellar cells
(b), whereais the solid phase with concentration
C
˜
S
a, andbthe solid phase with concentrationC
˜
S
b. The
average temperature at the interface is aroundT
Ibe-
low the eutectic temperatureT
E. The liquid at the
interface is at metastable concentrations (dashed
lines in (a)). The solid–liquid interfaces in (b) are
curved and they meet witha–binterfaces at three-
phase coexistence points.
(a) (b)www.iran-mavad.com
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206. solid–liquid interfaces relative to a flat
interface. Accordingly, the growth front
will have local curvature. What is worse,
there is no guarantee that thea–bsolid–
solid interfaces are really parallel to the
growth direction. Making a symmetry as-
sumption for simplicity again, we can
ignore this problem for the moment. The
curvatureKof the growth front at each la-
mella is then proportional to
l
–1
.Thiscur-
vature requires, through the Gibbs–Thom-
son relation, Eq. (2-102), another reduc-
tion,D
T
K, of the interface temperature be-
low
T
I:
D
T
K=T
EdK (2-132)
wheredis an effective capillary length, de-
pending on surface tensions, withKnow
taken asK~
l
–1
.
The total reduction of temperature below
T
Eduring eutectic growth can thus be writ-
ten as
(2-133)
with dimensionless constantsa
Ianda
K.
Plotting this supercooling as a function
of lamellar spacing, we find a minimal
supercooling∂D
T/∂l=0at
(2-134)
Again, this is the fundamental scaling rela-
tion conjectured in Eq. (2-1) and encoun-
tered in various places in this chapter,
where the origin is a competition between
driving force and stabilizing forces, most
intuitively expressed in Eq. (2-133).
A more thorough theoretical analysis of
eutectic growth was given in the seminal
paper by Jackson and Hunt (1966), which
is still a standard reference today. One ba-
sic approximation in this paper was to av-
erage the boundary conditions on flux and
temperature over the interface. This led to
Eq. (2-133), and it was argued that the min-
l=
KIaa ld/
DD DTT TTa
l
a
d
=+ =
IKEI K
l
l
+
⎧
⎨
⎩
⎫
⎬
⎭
imum undercooling would serve as the op-
erating point of the system with spacing
given by Eq. (2-134).
This stationary calculation was extended
by Datye and Langer (1981) to a dynamic
stability analysis, where the solid–solid–
liquid triple points could move parallel and
perpendicular to the local direction of
growth, coupled however to the normal di-
rection of the local orientation of the front.
It was found that the marginal stability co-
incided exactly with the point of minimum
undercooling.
The basic model equations for eutectic
growth in a thermal gradient field can be
written in scaled form as follows (Brattkus
et al., 1990; Caroli et al., 1990). Neglecting
diffusion in the solid (one-sided model)
and assuming a single diffusion coefficient
Dfor solute diffusion near eutectic concen-
trationC
E(Fig. 2-47), we assume a sym-
metric phase diagram aboutC
Efor simplic-
ity.
We define a relative (local) concentra-
tion gap as
(2-135)
withDC
a=C
E–C
˜
S
a(T),DC
b=C
˜
S
b(T)–C
E
andDC=DC
a+DC
b.
The dimensionsless composition is then
defined as
(2-136)
and (in contrast to the previous definitions)
we express lengths and times in units of a
diffusion lengthl˜and time
t˜
l˜=D/V,
t˜=D/V
2
(2-137)
(Note thatl˜here differs by a factor of 2
from previous definitions.) Restricting our
attention to two dimensions corresponding
to lamellar structures, we now define the
u
CC
CT
=
E
E−
D()
d=
D
D
aC
C
154 2 Solidificationwww.iran-mavad.com
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207. 2.6 Eutectic Growth 155
diffusion equation as
(2-138)
The conservation law at the interfacez=
z(x,t)is
–nˆ·∂u=D(x,t)(1+
z
·
)n
z (2-139)
with the unit vectornˆnormal to the inter-
face and
(2-140)
The Gibbs–Thomson relation for the
boundary becomes
(2-141)
(kinetic coefficients were considered by
Geilikman and Temkin (1984). Finally, and
this is a new condition in comparison with
the simple directional solidification, the
triple point where three interfaces meet
should be in mechanical equilibrium
g
aL+g
bL+g
ab= 0 (2-142)
with the surface tension vectors
goriented
so that each vector points out of the triple
point and is tangent to the corresponding
interface. As before,Kis the local curva-
ture of the interface, being positive for a
solid bulging into the liquid. The dimen-
sionless capillary and thermal lengths asso-
ciated with thea-phase are
(2-143)
(2-144)
l
l
mC
G
T
T
a
a
D
=
1
˜
()
d
l
T
mCL
a
a
aa
D
=
LE1
˜()gg
ut
dK
l
dK
l
T
T
(,)
/
/z
z
al
z
bl=
in -region
s
in -regions
−−
−−
⎧
⎨
⎪
⎪
⎩
⎪
⎪
a
a
b
b
D(,)
/
/
xt
x
x
=
for in -front regions
for in -front regions
da l
db l
−
⎧
⎨
⎩
1
∂
∂
∇+
∂
∂
u
t
u
z
u=
2
and are equivalent for thebphase, with
m
a=|dT/dC
˜
L
a|as the absolute liquidus
slope,G
Tas the fixed thermal gradient and
L
aas the specific latent heat. At infinity,
zƕ, the boundary condition to Eq. (2-
138) isu
•, depending on the initial concen-
tration in the liquid. This then forms a
closed set of equations.
Ahead of the eutectic front, the diffusion
field can be thought of as containing two
ingredients: a diffusion layer of thicknessl˜
associated with global solute rejection and
modulations due to the periodic structure
of the solid of the extent
l(l⎧l˜). When
the amplitude of the front deformations is
small compared to these lengths, the aver-
aging approximation by Jackson and Hunt
(1966) (and also by Datye and Langer,
1981) seems justified.
This point was taken up by Caroli et al.
(1990), who found that the approximation
is safe only in the limit of large thermal
gradients
G
T⎨Vm(DC)/D. (2-145)
This approximation appears difficult to
reach experimentally, though.
In an attempt to shed some light on
wavelength selection, Datye et al. (1981) and
Langer (1980b) considered finite ampli-
tude perturbations of the local wavelength
(Fig. 2-48). This type of approach was used
in a somewhat refined version by Brener et
al. (1987). They derived an approximate
potential function for wavelengths
land
argued that under finite amplitude of noise,
the wavelength selected on average is de-
fined by a balance in the creation rates and
the annihilation rates of lamellae. In other
words, if lamellae disappear through sup-
pression by neighboring lamellae and ap-
pear through nucleation, then an equal rate
of these processes leads to a selection of
an average spacing
l
–
, because both depend
on
l. The operating point was found in awww.iran-mavad.com
+ s e l ⎧'4 , kp e r i ⎧&s ! 9 j+ N 0 e

208. limited interval near the wavelength cor-
responding to minimal supercooling (or
maximal velocity in an isothermal process)
and, accordingly, is described by Eq. (2-
134).
More recent extensions of the theory
(Coullet et al., 1989) gave indications that
the orientation of the lamellae (under iso-
tropic material parameters) are necessarily
parallel to the growth direction of the front
but may be tilted and travel sideways at
some specific angles (Caroli et al., 1990;
Kassner and Misbah, 1991). Finally, it was
found (Kassner and Misbah, 1991b) that
the standard model of eutectic solidifica-
tion has an intrinsic scaling structure
(2-146)
with a scaling functionfdepending only on
l/l
Tso thatl~V
–1/2
for 2ll
Tor for high
enough velocities, while at lower velocities
the exponent should be smaller:
l~V
–0.3
.
This is in good agreement with the argu-
ments given for ordinary directional solid-
ification and also explains a lot of experi-
mental data (Lesoult, 1980).
Numerical simulations of the full contin-
uum model of eutectic growth were subse-
quently carried out by Kassner and Misbah
(1990, 1991a, b, c). They gave a fairly
complete description of the steady-state
structures to be expected in the eutectic
l~(/)ld f l l
T
system, revealing the similarities in the structure of solution space between eutec- tic and dendritic growth by showing the existence of a discrete set of solutions for each parameter set (Kassner and Mis- bah, 1991b). Axisymmetric solutions were shown to cease to exist beyond a certain wavelength by merging in a fold singular- ity. For larger wavelengths, extended tilted states appear. They were characterized in detail, both numerically (Kassner and Mis- bah, 1991c) and analytically (Misbah and Temkin, 1992; Valance et al., 1993). The supercritical nature of the tilt bifurcation was demonstrated and a prediction was made as to howextendedtilted domains
could be produced rather than the localized structures found so far. This prediction was borne out by subsequent experiments by the Faivre group (Faivre and Mergy, 1992a). Amplitude equations were derived that explained how a supercritical bifurca- tion could give rise to localized tilted do- mains and predicted a dynamical wave- length selection mechanism (Caroli et al., 1992; Kassner and Misbah, 1992a) as well as describing the influence of crystalline anisotropy (Kassner and Misbah, 1992a). Again, this was verified in detail experi- mentally (Faivre and Mergy, 1992b). A similarity equation was derived both for axisymmetric (Kassner and Misbah,
156 2 Solidification
Figure 2-48.Turbulent behav-
ior of lamellar spacings with
time in a simplified model of
eutectic growth (Datye et al.,
1981).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

209. 2.6 Eutectic Growth 157
1991b) and tilted states (Kassner and Mis-
bah, 1992b), giving the corrections to the
aforementioned scaling law. Anomalous
asymmetric cells bearing resemblance to
asymmetric cells in experiments on dilute
alloys (Jamgotchian et al., 1993) were
found in the numerical solution of the eu-
tectic growth problem and described ana-
lytically (Kassner et al., 1993).
To conclude the discussion of steady
states, it was shown numerically that a
number of periodicity-increasing bifurca-
tion exist, greatly enlarging the space of
possible stationary solutions (Baumann et
al., 1995a). These results strongly suggest
that the question of wavelength selection
should be considered from a different point
of view. It is known that a band of steady-
state solutions exists in directional solidifi-
cation beyond the instability threshold and
a similar statement holds for eutectic (even
though there is no threshold in this case).
Moreover, careful experiments (Faivre,
1996) have clarified that the observed
wavelength depends on the history of the
sample. Therefore, there is no wavelength
selection in the strict mathematical sense.
This is at least true in the absence of noise,
i.e. for the deterministic equations of mo-
tion. However, the investigations of Bau-
mann et al. (1995a) render it plausible that
a whole attractor of steady-state solutions
with accumulation points exists. One of
these accumulation points is the point of
marginal stability. Since the temporal evo-
lution of any structure in the vicinity of this
attractor will become slow, it seems likely
that observed wavelengths will correspond
to points of increased density of this attrac-
tor. There have been objections (Karma
and Rappel, 1994) that many of the newly
found steady-state solutions are on the low-
wavelength side of the point of marginal
stability, hence they are unstable and there-
fore irrelevant for pattern formation. This
objection was countered (Baumann et al.,
1995b) by pointing out that there is quite a
number of steady states on the high-wave-
length side of this point and that the ques-
tion of irrelevance for pattern formation is
a question of time-scales. Since these be-
come slow near the accumulation point, the
relevance of a density of additional states
cannot be easily discarded. It should be
kept in mind that tilted lamellar states are
also unstable with respect to phase diffu-
sion near the tilt bifurcation (Fauve et al.,
1991). Nevertheless, they survive experi-
mental time-scales, because phase diffu-
sion is very slow.
The firstdynamicsimulation of the con-
tinuum model of eutectic growth (in the
quasi-stationary approximation) appears to
have been done by Kassner et al. (1995),
but the only extensive study so far is due to
Karma and Sarkissian (1996). This beauti-
ful work was done in close collaboration
with experimentalists and thus validated
quantitively. It pinpointed a number of new
short-wavelength oscillatory instabilities
and made quantitative predictions for the
CBr
4–C
2Cl
6system allowing these in-
stabilities to be reproduced in experiments
(Faivre, 1996; Ginibre et al., 1997). There-
fore, a pretty complete picture exists today
of the possible stationary and oscillatory
patterns in lamellar eutectic growth and
their stability range.
A recent development is the stability
analysis of ternary eutectics by Plapp and
Karma (1999), based on an extension of the
Datye–Langer theory. It shows how the
presence of a third component in the sys-
tem leads to the formation of eutectic colo-
nies; an effective description of the eutectic
front on length scales much larger than the
lamellar spacing is provided, which yields
a simple means of calculating an approxi-
mate stability spectrum.www.iran-mavad.com
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210. 2.6.2 Experimental Results
on Eutectic Growth
Eutectic growth was a subject of much
interest to experimentalists in the late
1960s and early 1970s. Substantial re-
search has been motivated by the possibil-
ity of developing new high-temperature
materials. Thein situdirectional solidifica-
tion of two phases of very different proper-
ties is an interesting method of producing
composite materials with exceptional prop-
erties. However, since these materials could
not outperform the more conventional, di-
rectionally solidified dendritic superalloys
in the harsh environment of a gas turbine,
the interest dropped. Therefore, most of the
research on eutectics was performed before
1980 (for a review, see Kurz and Sahm,
1975; Elliott, 1983). One exception is the
ongoing research concerning casting of eu-
tectic alloys such as cast iron (Fe–C or
Fe–Fe
3C eutectic) and Al–Si.
Casting alloys are generally inoculated
and solidify in equiaxed form (free eutectic
growth) (see Flemings, 1991). This fact,
however, does not make any substantial
difference to their growth behavior because
growth is solute diffusion controlled in
nearly all cases owing to the high concen-
tration of the second element. The models
described above therefore apply to both di-
rectional and free solidification.
The different alloys can be classified
into four groups of materials (Kurz and
Fisher, 1998): lamellar or fibrous systems,
and non-faceted/non-faceted (nf/nf) or
non-faceted/faceted (nf/f) systems.
The distinction between f and nf growth
behavior can be made with the aid of the
melting entropy. Small entropy differences
DS
fbetween liquid and solid (typical for
metals and plastic crystals such as SCN,
PVA, etc.) lead to nf growth with atomi-
cally rough interfaces. Materials with large
DS
fvalues are prone to form atomically
smooth interfaces, which lead to the forma-
tion of macroscopically faceted appear-
ance.
In the case of nf/nf eutectics, volume
fractions (of one eutectic phase) of less
than 0.3 lead generally to fibers, while
at volume fractions between 0.3 and 0.5,
lamellar structures prevail. The micro-
structures of nf/nf eutectics (often simple
metal/metal systems) are considered regu-
lar and those of nf/f eutectics (mostly the
above-mentioned casting alloys) are con-
sidered irregular. Fig. 2-49 shows schemat-
ically the morphology of the growth front
in both cases. It can be easily understood
that growth in nf/nf eutectics is much more
of a steady-state type than it is in f/nf eu-
tectics.
Regular structures
Applying a criterion such as growth at
the extremum to the solution of the capil-
158 2 Solidification
Figure 2-49.Solid/liquid interface morphologies of
(a) regular (nf/nf) eutectics and of (b) irregular (nf/f)
eutectics during growth.www.iran-mavad.com
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211. 2.6 Eutectic Growth 159
lary-corrected diffusion equations (Jackson
and Hunt, 1966), Eq. (2-133), we obtain for
nf/nf eutectics the well-known relation-
ships (Eq. (2-134))
l
2
V=C
DT
2
/V=4C¢
whereCandC¢are constants. Fig. 2-50
shows that this behavior has been observed
globally in many eutectic systems, some of
them having been studied over five orders
of magnitude in velocity. The situation is
much less clear when it comes to analyzing
eutectoid systems. (Eutectoids are “eutec-
tics” with the liquid parent phase replaced
by a solid.) Often a
l
3
Vrelationship is
found in these systems (Fig. 2-50b) over
some range of the variables (see Eq. (2-
146)).
In general, it may be said that the field of
eutectic growth is under-represented in ma-
terials research, and many more careful
studies are needed before a clearer insight
into their growth can be gained. Trivedi
and coworkers began such research in the
early 1990s, and some of their results are
Figure 2-50.Lamellar spacings of (a) directionally solidified eutectics and (b) directionally transformed eutec-
toids as a function of growth rate (Kurz and Sahm, 1975).
(a)
(b)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

212. presented here. Fig. 2-51 indicates that in
eutectics the spacings are also not at all
uniquely defined. There is a rather wide
distribution around a mean value for each
rate (Trivedi et al., 1991). The operating
range of eutectics is determined by the per-
manent creation and movement of faults
(see below). This process is three-dimen-
sional and cannot be realistically simulated
in two-dimensional calculations.
In Fig. 2-52, the mean spacings are plot-
ted as points, and the limits of the distribu-
160 2 Solidification
Figure 2-51.Eutectic spacing distribution curves as a function of velocity for Pb–Pd eutectics (Trivedi et al.,
1991).
Figure 2-52.A comparison of the ex-
perimental results on the interlamellar
spacing variation with velocity for
CBr
4–C
2Cl
6with the theoretical values
(solid lines) for two marginally stable
spacings (Seetharaman and Trivedi,
1988).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

213. 2.6 Eutectic Growth 161
tion are given by the extension of the
bars. From the calculated range of stability,
which was discussed by Jackson and Hunt
(1966), it can be seen that the minimum
of the experimental values coincides with
the theoretical prediction (see also Sec.
2.6.1). This, however, does not provide
definitive proof of this prediction, due to
the fact that several physical parameters
of the system are not precisely known. On
the other hand, it is clear that eutectic spac-
ings do not explore the upper range of
stability (catastrophic breakdown), at least
not in nf/nf systems. Some other mecha-
nism limits the spacing at its upper bound.
The adjustment of spacings is a rapid pro-
cess, and its rate increases when the spac-
ing increases (Seetharaman and Trivedi,
1988).
Irregular structures
The above relationship for eutectic spac-
ing and undercooling as a function of
growth rate are also useful in the case of ir-
regular systems such as the nf/f casting al-
loys Fe–C or Al–Si (Fig. 5-49b). Jones
and Kurz (1981) introduced a factor,
j,
which is equal to the ratio of the mean
spacing,·
lÒ, of the irregular structure to
the spacing at the extremum. This leads to
the following relationships:
·
lÒ
2
V=j
2
C
·DTÒ
2
/V=[j+(1/j)]
2
C¢
Faults
Defects in the ideal lamellar or fibrous
structure are an essential ingredient of eu-
Figure 2-53.Eutectic fault struc-
tures in directionally solidified
Al–CuAl
2alloy (Double, 1973).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

214. tectic growth. They allow the two-phase
crystal to rotate into crystallographically
(energetically) favorable orientations (Ho-
gan et al., 1971) and to adapt its spacing to
the local growth conditions. In lamellar
systems, there are different types of faults
(Double, 1973): single or extended faults,
with or without net mismatch (Fig. 2-53).
They mostly represent sub-boundaries of a
eutectic grain and could develop through
polygonization of dislocations which form
because of the stresses created at the inter-
phase boundaries of the composite. Fig. 2-
54 shows distributions of fault spacings,
L,
for different growth rates indicated by dif-
ferent lamellar spacings,
l(Riquet and Du-
rand, 1975). In the case of non-faceted fi-
brous structures, the faults are formed by
simple fiber branches or terminations.
Oscillations
Periodic oscillations have been observed
as a morphological instability in several
systems grown under various conditions
(Hunt, 1987; Carpay, 1972; Zimmermann
et al., 1990; Gill and Kurz, 1993, 1995).
These morphological instabilities form in
off-eutectic alloys even at growth rates
of several cm/s, as is shown in Fig. 2-55.
162 2 Solidification
Figure 2-54.Eutectic fault spacing distribution
curves for Al–CuAl
2directionally solidified with
different growth rates and spacing values as indi-
cated (Riquet and Durand, 1975).
Figure 2-55.Periodic oscillations in hypoeutectic
Al–CuAl
2eutectic under rapid laser resolidification
conditions: (a) experimental observation and (b) sim- ulation (Zimmermann et al., 1990; Karma, 1987).
(a)
(b)www.iran-mavad.com
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215. 2.6 Eutectic Growth 163
There is good correspondence of the ob-
served structures with the results of theo-
retical modeling by Datye and Langer
(1981) and by Karma (1987). In recent ex-
periments a complete stability diagram for
the lamellar CBr
4–C
2Cl
6eutectic has been
determined by Ginibre et al. (1997) (Fig.
2-56a). It shows the observed structures
for the variables dimensionless spacing
(
L) and concentration (h). In the limits
1<
L< 2 the basic state is stable, but over
a decreasing composition range when
L=l/l
minis increased. Outside this re-
gime, three different states and combina-
tions of these have been found; tilt mode
(T), 1
loscillations (period-preserving os-
cillatory or optical mode) and 2
loscilla-
tions (period-doubling oscillatory mode,
Fig. 2-56). These results are in good quan-
titative agreement with numerical results
by Karma and Sarkissian (1996).
As was the case with dendrites, eutectics
also form extremely fine microstructures
when they are subject to rapid solidifica-
tion. A value of
l=15 nm (each phase is
some 20 atoms wide) seems to be the mini-
mum spacing that can be achieved with
growth rates of the order of 0.2 m/s (Zim-
mermann et al., 1989). In highly under-
cooled alloys anomalous eutectic struc-
tures have been found which, as with den-
drites, seem to be the result of the large
capillary forces involved with such fine
scales (Goetzinger et al., 1998).
2.6.3 Other Topics
Peritectics
A wide spectrum of microstructures can
be found in peritectic alloys. Recently this
has produced a great deal of interest (Kerr
and Kurz, 1996). In directional solidifica-
tion, when theG/Vratio is high enough,
i.e., in the range of the limit of constitu-
tional undercooling of the phase with the
smaller distribution coefficient, more com-
plex microstructures can form in the two-
phase region. New phases may appear dur-
ing transient or steady-state growth via nu-
cleation at the solidification front. The ac-
Figure 2-56.Stability diagram of directionally so-
lidified CBr
4–C
2Cl
6alloys. (a) Symbols: experimen-
tally observed periodic patterns (Ginibre et al.,
1997). Lam. Term.: lamellar termination. (b) Curves:
numerically calculated diagram (Karma and Sarkis-
sian, 1996).
(a)
(b)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

216. tual microstructure selection process is
thus controlled by nucleation of the phases,
and by growth competition between the nu-
cleated grains and the pre-existing phase
under non-steady-state conditions. In this
case nucleation in the constitutionally
undercooled zone ahead of the growth
front has to be taken into account in order
to determine the microstructure selection
(Ha and Hunt, 1997; Hunziker et al., 1998).
Further, new structures have been observed
which are controlled by convection. More
detailed discussions of this subject can be
found in Karma et al. (1998), Park and Tri-
vedi (1998), Trivedi et al. (1998), Van-
dyoussefi et al. (2000), Mazumder et al.
(2000).
Phase/microstructure selection
Phase and microstructure selection is of
utmost importance to applications of solid-
ification theory as it controls the properties
of materials. Nucleation and growth have
to be modeled in order to make predictions
of structures as a function of composition
and the solidification variables; growth
rate and temperature gradient for direc-
tional solidification, undercooling for free
growth. The interested reader is referred to a
recent overview by Boettinger et al. (2000)
where these phenomena are discussed.
Micro/macro modeling
The last decade has experienced signifi-
cant progress in numerical modeling of
combined macroscopic and microscopic
solidification phenomena. Nucleation and
growth models have been implemented in
2D and 3D heat flow and fluid flow pro-
grams providing a substantial improvement
of our analytical tools for optimized and
new materials processes such as single
crystal casting of superalloys for gas tur-
bines. The interested reader is referred to
Rappaz (1989), Rappaz et al. (2000), Wang
et al. (1995), Wang and Beckermann (1996),
Beckermann and Wang (1996), Boettinger
et al. (2000).
2.7 Summary and Outlook
During the recent years, very substantial
improvements in our understanding of the
pattern-forming processes in solidification
have been achieved. Although the basic
model equations have been known for sev-
eral decades, it is only during recent years
that the mathematical and numerical tools
were extended to allow for a reliable analy-
sis of the complicated expressions. In addi-
tion, careful experiments have been per-
formed, mostly on model substances, which
have provided an impressive amount of
precise, quantitative data. This combined
effort has basically solved the problem of
free dendritic growth with respect to veloc-
ity selection and side branch formation.
In the process of directional solidifica-
tion, a consistent picture is now emerging,
relating the growth mode to free dendritic
growth and, at the same time, to viscous
fingering and growth in a channel. At very
high growth rates that approach the limit
of absolute stability, the situation is still
somewhat unclear, for non-equilibrium
effects like kinetic coefficients then be-
come of central importance. These quan-
tities are difficult to determine experimen-
tally. Furthermore, the selection of the pri-
mary spacings of the growing array of cells
and dendrites is still subject to discussion.
One such point of contention, for example,
is the typically observed increase in spac-
ing when moving from cellular to dendritic
structures in model substances.
In eutectic growth, the situation is even
less understood, for good reason. The
164 2 Solidificationwww.iran-mavad.com
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217. 2.8 References 165
three-phase junctions at the solid–liquid
interface enter as additional conditions and
further details of the phase diagram be-
come important. The dynamics of the sys-
tem seem to show a richer structure than
ordinary directional solidification. The se-
lection of spacings between the different
solid phases in materials of practical im-
portance occurs through three-dimensional
defect formation. In addition, nucleation
and faceting of the interfaces should be
considered.
A number of problems common to all
of these growth modes have only been
touched upon so far. These problems in-
clude, for example, the redistribution of
material far behind the tip regions, the
treatment of elastic effects, and the interac-
tion with hydrodynamic instabilities due to
thermal and compositional gradients.
In summary, we expect the field to re-
main very active in the future, as it is at-
tractive from a technological point of view.
It will certainly provide some surprises and
new insights into the general concepts of
pattern formation in dissipative systems.
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170 2 Solidificationwww.iran-mavad.com
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223. 3 Diffusion Kinetics in Solids
Graeme E. Murch
Department of Mechanical Engineering, University of Newcastle, N.S.W. 2308, Australia
List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.1 Introduction................................. 175
3.2 Macroscopic Diffusion........................... 175
3.2.1 Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
3.2.2 Types of Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . 178
3.2.2.1 Tracer or Self-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
3.2.2.2 Impurity and Solute Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 178
3.2.2.3 Chemical or Interdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 178
3.2.2.4 Intrinsic or Partial Diffusion Coefficients . . . . . . . . . . . . . . . . . . 179
3.2.2.5 Surface Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 180
3.2.3 Phenomenological Equations of Irreversible Thermodynamics . . . . . . . 181
3.2.3.1 Tracer Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.2.3.2 Chemical Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
3.2.3.3 Einsteinian Expressions for the Phenomenological Coefficients . . . . . . 185
3.2.3.4 Relating Phenomenological Coefficients to Tracer Diffusion Coefficients . 185
3.2.4 Short-Circuit Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
3.3 Microscopic Diffusion........................... 189
3.3.1 Random Walk Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.3.1.1 Mechanisms of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.3.1.2 The Einstein Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
3.3.1.3 Tracer Correlation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 193
3.3.1.4 Impurity Correlation Factor . . . . . . . . . . . . . . . . . . . . . . . . . 196
3.3.1.5 Correlation Factors for Concentrated Alloy Systems . . . . . . . . . . . . 199
3.3.1.6 Correlation Factors for Highly Defective Systems . . . . . . . . . . . . . 201
3.3.1.7 The Physical or Conductivity Correlation Factor . . . . . . . . . . . . . . 204
3.3.1.8 Correlation Functions (Collective Correlation Factors) . . . . . . . . . . . 206
3.3.2 The Nernst–Einstein Equation and the Haven Ratio . . . . . . . . . . . . 211
3.3.3 The Isotope Effect in Diffusion . . . . . . . . . . . . . . . . . . . . . . . 213
3.3.4 The Jump Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
3.3.4.1 The Exchange Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 217
3.3.4.2 Vacancy Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
3.4 Diffusion in Materials........................... 219
3.4.1 Diffusion in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
3.4.1.1 Self-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
3.4.1.2 Impurity Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
3.4.2 Diffusion in Dilute Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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224. 3.4.2.1 Substitutional Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
3.4.2.2 Interstitial Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
3.4.3 Diffusion in Concentrated Binary Substitional Alloys . . . . . . . . . . . 226
3.4.4 Diffusion in Ionic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 227
3.4.4.1 Defects in Ionic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
3.4.4.2 Diffusion Theory in Ionic Crystals . . . . . . . . . . . . . . . . . . . . . 228
3.5 Experimental Methods for Measuring Diffusion Coefficients...... 231
3.5.1 Tracer Diffusion Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 231
3.5.2 Chemical Diffusion Methods . . . . . . . . . . . . . . . . . . . . . . . . 232
3.5.3 Diffusion Coefficients by Indirect Methods . . . . . . . . . . . . . . . . . 233
3.5.3.1 Relaxation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
3.5.3.2 Nuclear Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
3.5.4 Surface Diffusion Methods . . . . . . . . . . . . . . . . . . . . . . . . . 234
3.6 Acknowledgements............................. 235
3.7 References.................................. 235
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225. List of Symbols and Abbreviations 173
List of Symbols and Abbreviations
a lattice parameter
b
1, b
2 solvent enhancement factors
B
1, B
2 solute enhancement factors
C, C
i concentration: particles per unit volume of species i
c
A,c
B,c
i,c
v,c
pmole fractions of metal A, B, species i, vacancies, paired interstitials
D diffusion coefficient (in m
2
s
–1
)
D¢ short-circuit diffusion coefficient
D˜ collective diffusion coefficient or interdiffusion coefficient
D* tracer diffusion coefficient
D
I
A
, D
I
B
, etc. intrinsic or partial diffusion coefficient of metal A, B, etc.
D
l lattice diffusion coefficient
D
s diffusion coefficient derived from the ionic conductivity
D
0
pre-exponential factor
e electronic charge
E
v
f energy of vacancy formation
E
b, E
B binding energy of impurity to dislocation (impurity-vacancy binding energy)
f, f
I(f
c) tracer correlation factor, physical correlation factor
F
v
f Helmholtz free energy for vacancy formation
g coordination
G
m
Gibbs free energy of migration
H
v
f enthalpy of vacancy formation
H
m
enthalpy of migration
H
R Haven ratio
J flux of atoms
k Boltzmann constant
K equilibrium constant
L phenomenological coefficients
l spacing between grain boundaries (average grain diameter)
l distance between dislocation pinning points
m mass
P
–
AV vacancy availability factor
Q activation energy for diffusion
R vector displacement
DR
i total displacement of species i
r, r
q jump distance of atom, of charge q
S
v
f entropy of vacancy formation
S
m
entropy of migration
t time
T temperature
T
M.Pt. temperature of melting point
u mobility
·vÒ average drift velocity
V volumewww.iran-mavad.com
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226. x distance, coordinate
X
i, X
v driving force on species i, vacancies
Z
i number of charges on i
g activity coefficient
G, G
q, G
i jump frequency (of charge q ; of species i)
m chemical potential
n vibration frequency
s ionic conductivity
w exchange frequency
CASCADE computer code
DEVIL computer code
erf Gaussian error function
PPM path probability method
SIMS secondary ion mass spectrometry
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227. 3.2 Macroscopic Diffusion 175
“The elementary diffusion process is
so very fundamental and ubiquitous
in the art and science of dealing with
matter in its condensed phase that it
never ceases to be useful but, at the
same time, is a problem which is
never really solved. It remains impor-
tant by any measure.”
D. Lazarus, 1984
3.1 Introduction
Many phenomena in materials science
depend in some way on diffusion. Common
examples are sintering, oxidation, creep,
precipitation, solid-state chemical reac-
tions, phase transformations, and crystal
growth. Even thermodynamic properties
and structure are sometimes dependent on
diffusion, or rather the lack of it. Many of
these phenomena are the subjects of other
contributions in this series. This contribu-
tion is concerned with the fundamentals of
the diffusion process itself.
The depth of subject matter is generally
introductory, and no prior knowledge of
solid-state diffusion is assumed. Where
possible, the reader is directed to more de-
tailed texts, reviews, and data compila-
tions.
3.2 Macroscopic Diffusion
3.2.1 Fick’s Laws
Although diffusion of atoms, or atomic
migration, is always occurring in solids at
temperatures above absolute zero, for mac-
roscopically measurable diffusion a gradi-
ent of concentration is required. In the
presence of such a concentration gradient
∂C/∂x(where C is the concentration in, say,
particles per unit volume and xis the dis-
tance) in one direction of a certain species
of atom, a flux Jof atoms of the same spe-
cies is established down the concentration
gradient. The law relating flux and concen-
tration gradient is Fick’s First Law, which,
for an isotropic medium or cubic crystal
can be expressed as
(3-1)
The proportionality factor or “coefficient”
of ∂C/∂xis termed the “diffusion coeffi-
cient” or less commonly the “diffusivity”.
The recommended SI units for Dare m
2
s
–1
but much of the literature is still in the
older c.g.s. units cm
2
s
–1
. The negative sign
in Eq. (3-1) arises because the flux is in the
opposite direction to the concentration gra-
dient. This negative sign could, of course,
have been absorbed into D, but it is more
convenient for Dto be a positive quantity.
When an external force such as an
electric field also acts on the system a more
general expression can be given:
(3-2)
where ·vÒis the average velocity of the
center of mass arising from the external
force on the particles. The first term in Eq.
(3-2) is thus the diffusive term, and the sec-
ond term is the drift term. Note the inde-
pendence of these terms. The external force
here is assumed to be applied gradually so
that the system moves through a series of
equilibrium states. When the force is sud-
denly applied the system can be thrown out
of equilibrium. These matters and the types
of external force are considered by Flynn
(1972) in a general consideration of diffu-
sion under stress.
By itself, Fick’s First Law (Eq. (3-1)) is
not particularly useful for diffusion mea-
surements in the solid state since it is virtu-
JD
C
x
C=−
∂
∂
+〈 〉v
JD
C
x
=−
∂
∂www.iran-mavad.com
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228. ally impossible to measure an atomic flux
unless steady state is reached. Further, be-
cause solid-state diffusivities are generally
small, the attainment of steady state in
a macroscopic specimen can take a very
long time. In a few cases, where the solid-
state diffusivity is high, for example car-
bon diffusion in austenite (Smith, 1953),
the steady-state flux and the concentration
gradient can be measured and the diffusiv-
ity obtained directly from Eq. (3-1).
In order to produce a basis for measuring
the diffusion coefficient, Eq. (3-1) is usu-
ally combined with the equation of con-
tinuity:
(3-3)
to give Fick’s Second Law:
(3-4)
If the diffusion coefficient is independent
of concentration and therefore position,
then Eq. (3-4) reduces to
(3-5)
The second-order partial differential equa-
tion Eq. (3-5) (or (3-4)) is sometiems called
the “diffusion equation”. Eq. (3-2) can also
be developed in the same way to give
(3-6)
and, if ·vÒand Dare independent of C, then
(3-7)
In order to obtain a solution to the diffu-
sion equation, it is necessary to establish
the initial and boundary conditions. Once a
∂
∂
∂
∂
−〈 〉
∂
∂
C
t
D
C
x
C
x
=
2
2
v
∂
∂
∂
∂
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟−
∂
∂
〈〉
C
tx
D
C
xx
C=( )v
∂
∂
∂
∂
C
t
D
C
x
=
2
2
∂
∂
∂
∂
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
C
tx
D
C
x
=
∂
∂
−
∂
∂
J
x
C
t
=
solution, C(x,t), of the diffusion equation
has been established, the diffusion coeffi-
cient itself is obtained as a parameter by
fitting the experimental C(x,t) to the ana-
lytical C(x,t). In the following we shall fo-
cus on some analytical solutions for some
well-known initial and boundary condi-
tions used in experimental diffusion stud-
ies.
Let us first examine some solutions for
Eq. (3-5). In a very common experimental
arrangement for “self” and impurity diffu-
sion a very thin deposit of amount Mof ra-
dioactive isotope is deposited as a sand-
wich layer between two identical samples
of “infinite” thickness. After diffusion for a
time tthe concentration is described by
(3-8)
which is illustrated in Fig. 3-1. If the de-
posit is left as a surface layer rather than a
sandwich, C(x,t) is doubled.
In another experimental arrangement,
the surface concentration C
sof the diffus-
ing species is maintained constant for time
t, perhaps by being exposed to an atmo-
sphere of it. Again, the substrate is thick or
mathematically infinite. The solution of the
diffusion equation is
(3-9)
where C
0is the initial or background con-
centration of the diffusing species in the
substrate and erf is the Gaussian error func-
tion defined by
(3-10)
This function is now frequently available
as a scientific library function on most
modern computers. Accurate series expan-
erf = dzuu
z
2
0
2
p
∫ −exp ( )
Cxt C
CC
xDt
(,)
(/ )
−
−
s
s
= erf
0
2
Cxt
M
Dt
xDt( , ) exp ( / )=
2
4
2
p
−
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229. 3.2 Macroscopic Diffusion 177
sions can be found in mathematical func-
tion handbooks.
The total amount of diffusant Staken up
(or, in fact, lost, depending on the relative
values of C
sand C(x,t)) from the substrate
is given by
S(t) = [2/C
0–C
s)]/A(Dt/p)
1/2
(3-11)
where A is the surface area of the sample.
A further common experimental ar-
rangement is the juxtaposition of two “infi-
nite” samples, one of which has a uniform
concentration C
0and the other a concentra-
tion C
1of the diffusing species. After time
tthe solution is
(3-12)
This gives a time evolution of the concen-
tration profile as shown in Fig. 3-2.
On many occasions where this experi-
mental arrangement is used, we are con-
cerned with diffusion in a chemical compo-
sition gradient. The relevant diffusion co-
efficient (see Sec. 3.2.2.3) is frequently de-
pendent on concentration, so Eq. (3-12) is
then inappropriate, and a solution to Eq.
(3-4) where D=D(C) must be sought. A
wellknown technique for this is the graphi-
cal integration method, usually called the
Boltzmann–Matano analysis. The general
solution for the concentration-dependent
Cxt C
CC
xDt
(,)
(/ )]
−
−
−
0
101
2
12= [ erf
diffusion coefficient can be expressed as
(Boltzmann, 1894; Matano, 1933)
(3-13)
where
(3-14)
A very detailed description of the use of
this analysis is given by Borg and Dienes
(1988).
In many practical situations of interdif-
fusion the relevant phase diagram traversed
will ensure that diphasic regions or new
phases will appear. This does not imply in-
cidentally that allequilibrium phases will
thus appear. The Boltzmann–Matano anal-
ysis can still be applied within single phase
regions of the concentration profile. The
growth of a new phase, provided it is dif-
fusion controlled, can usually be described
by a parabolic time law. These matters are
dealt with in detail by Philibert (1991).
The solutions of the diffusion equation
given here are among the more commonly
encountered ones in solid-state diffusion
studies. Numerous others have been given
by Carslaw and Jaeger (1959) and Crank
(1975).
C
C
xC
1
0
0∫d=
DC
t
x
C
xC
CC
C
()′−
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
′
′
∫=d
1
2
1
Figure 3-1.Time evolution of Eq. (3-8). Figure 3-2.Time evolution of Eq. (3-12).www.iran-mavad.com
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230. 3.2.2 Types of Diffusion Coefficients
For the non-specialist, the meaning and
significance of the various diffusion coeffi-
cients used can be confusing because of in-
consistent terminology in the literature. An
attempt will be made here to clarify the sit-
uation as much as possible.
3.2.2.1 Tracer or Self-Diffusion
Consider a well-annealed sample of a
pure metal. Although atoms are diffusing
about in the sample at a rate depending on
temperature, from a macroscopic point of
view nothing appears to be happening. In
order to observe diffusion macroscopically
we must impose a concentration gradient.
For the case of a pure metal, a radioactive
tracer of the same metal is used. The result-
ing diffusion coefficient is termed the
tracer diffusion coefficientwith the symbol
D*. Because the tracer is chemically the
same as the host, this diffusion coefficient
is also termed the self-diffusion coefficient,
although sometimes this terminology is re-
served for the tracer diffusion coefficient
divided by the tracer correlation factor, f
(see Sec. 3.3.1.3). The same idea is readily
extended to alloys and compounds. How-
ever, care must always be taken that a
chemical composition gradient is not un-
wittingly imposed. For example, in order to
measure the oxygen self-diffusion coeffi-
cient in a nonstoichiometric compound
such as UO
2+x, we can permit
18
O (a stable
isotope that can be probed later by nuclear
analysis or Secondary Ion Mass Spectrom-
etry) to diffuse in from the gas phase, pro-
vided that the partial pressure of oxygen is
already in chemical equilibrium with the
composition of the sample. Alternatively, a
layer of U
18
O
2+xcan be deposited on the
surface of a sample of UO
2+xprovided that
it has exactly the same chemical composi-
tion as the substrate.
3.2.2.2 Impurity and Solute Diffusion
In order to measure the impurity diffu-
sion coefficient, the tracer is now the im-
purity and is different chemically from the
host. However, the concentration of impur-
ity must be sufficiently low that there is not
a chemical composition gradient. Strictly,
of course, the tracer impurity should be
permitted to diffuse into the sample already
containing the same concentration of im-
purity. In practice, the concentration of
tracer impurity is normally kept extremely
low, thereby making this step unnecessary.
Because the impurity is always in stable
solid solution (unless implanted), is often
termed the solute and the impurity diffu-
sion coefficient is sometimes also termed
the solute diffusion coefficient at infinite
dilution. However, the terminology solute
diffusion coefficient is often “reserved” for
dilute alloys, where, in addition, we often
measure the solvent diffusion coefficient.
In the context of those experiments both
solute and solvent diffusion coefficients
frequently depend on solute content (see
Sec. 3.4.2). In all of these experiments, as
in self-diffusion, the chemical composition
of the sample must remain essentially un-
changed by the diffusion process, other-
wise it is a chemical diffusion experiment.
3.2.2.3 Chemical or Interdiffusion
So far, we have discussed diffusion coef-
ficients which are measured in the absence
of chemical composition gradients. Chemi-
cal diffusion is the process where diffusion
takes place in the presence of a chemical
composition gradient. It is the diffusion co-
efficients describing this process which
generate the greatest amount of confusion.
It is helpful to look at several examples.
Consider first diffusion in a pseudo-one-
component system. One example is the dif-
fusion of an adsorbed monolayer onto a
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231. 3.2 Macroscopic Diffusion 179
clean section of surface. Another is the dif-
fusion between two metal samples differ-
ing only in their relative concentrations of
a highly mobile interstitial species such as
H. A further example is diffusion between
two nonstoichiometric compounds, e.g.,
Fe
1–dO and Fe
1–d¢O. In all of these cases
only onespecies of atom is involved in the
diffusion process which brings the system
to a common composition (which is why it
can be considered a pseudo-one-compo-
nent system). Often the process can be pic-
tured as the interdiffusion of vacant sites
and atoms. The diffusion coefficient de-
scribing this process is usually called the
chemical diffusion coefficient, sometimes
the interdiffusion coefficient, and occasion-
ally, the collective diffusion coefficient.
Generally the symbol used is D˜. For these
pseudo-one-component systems, the pre-
ferred name is chemical diffusion coeffi-
cient. In general, the chemical diffusion co-
efficient does not equal the self-diffusion
coefficient because of effects arising from
the gradient of chemical composition, see
Eq. (3-99).
Chemical diffusion in binary substitu-
tional solid solutions is frequently called
interdiffusion.In a typical case pure metal
A is bonded to pure metal B and diffusion
is permitted at high temperature. Although
bothA and B atoms move, only one con-
centration profile, say of A, is established
(the profile from B contains no new infor-
mation). The resulting diffusion coefficient
which is extracted from the profile, by the
Boltzmann–Matano analysis (e.g., see Sec.
3.2.1) is termed the interdiffusion coeffi-
cient and is given the symbol D˜. Not infre-
quently, the diffusion coefficient for this
situation is loosely called the chemical dif-
fusion coefficient or the mutual diffusion
coefficient. This single diffusion coeffi-
cient is sufficient to describe the concentra-
tion profile changes of the couple. Because
of its practical significance the interdiffu-
sion coefficient is the one often quoted in
metal property data books (Brandes, 1983;
Mehrer, 1990; Beke 1998, 1999). See also
the following section for further discussion
of D˜.
3.2.2.4 Intrinsic or Partial Diffusion
Coefficients
In contrast to the pseudo-one-component
systems described above where the dif-
fusion rates of the atoms and vacant sites
are necessarily equal, in the substitutional
binary alloy the individual diffusion rates
of A and B are not generally equal since the
corresponding self-diffusion coefficients
are not. In the interdiffusion experiment
this implies that there is a net flux of atoms
across any lattice plane normal to the diffu-
sion direction. If the number of lattice sites
is conserved, each plane in the diffusion re-
gion must then shift to compensate. This
shift with respect to parts of the sample
outsidethe diffusion zone, say the ends of
the sample, is called the Kirkendall effect.
This effect can be measured by observing
the migration of inert markers, usually fine
insoluble wires which have been incorporat-
ed into the sample before the experiment.
The assumption is that the wires follow the
motion of the lattice in their vicinity.
The intrinsic diffusion coefficients of A
and B, D
I
A
and D
I
B
, are defined with refer-
ence to the fluxes of A and B relative to the
locallattice planes:
(3-15)
and
(3-16)
The diffusion coefficients are sometimes
also termed partial diffusion coefficients.
′−
∂
∂
JD
C
xBB
I B=
′−
∂
∂
JD
C
xAA
I A=www.iran-mavad.com
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232. If vis the velocity (Kirkendall velocity)
of the lattice plane measured with respect
to parts of the sample outside the diffusion
zone then the fluxes with respect to it are
J
A=J
A¢+vC
A (3-17)
and
J
B=J
B¢+vC
B (3-18)
where C
Aand C
Bare the concentrations at
the lattice planes. It should be noted that
the interdiffusion coefficient is measured
in this frame of reference. Since ∂C
A/∂x
=–∂C
B/∂x, we easily deduce that
(3-19)
and
(3-20)
where c
Aand c
Bare the mole fractions of
A and B. The term in parentheses in Eq.
(3-20) is the interdiffusion coefficient D ˜:
D˜=c
AD
I
B
+ c
BD
I
A
(3-21)
The interdiffusion coefficient is seen to be
a weighted average of the individual (in-
trinsic) diffusion coefficients of A and B.
With the aid of Eqs. (3-19) and (3-21) the
intrinsic diffusion coefficients can be de-
termined if D˜and vare known. The compo-
sition they refer to is the composition at the
inert marker. Note that if there is nomarker
shift, Eq. (3-19) implies that the intrinsic
coefficents are equal. The relationship of
the intrinsic diffusion coefficient to the
tracer (self) diffusion coefficient is ex-
plored in Sec. 3.2.3.2.
From the form of Eq. (3-21) when
c
AÆ0, it can be seen that D˜ÆD
I
A
. It
should also be noted that in this limit D
I
A
reduces to the impurity diffusion coeffi-
cient of A in B.
JcDcD
C
x
AAB
I
BA
I A=−+
∂
∂
()
v=
A
I
B
I A()DD
C
x
−
∂
∂
3.2.2.5 Surface Diffusion Coefficients
Surface diffusion refers to the motion of
atoms, sometimes molecules, over the sur- face of some substrate. The diffusing spe- cies can be adsorbed atoms, e.g. impurity metal atoms on a metal substrate (this is normally referred to as hetero-diffusion) or of the same species as the substrate (this is usually referred to as self-diffusion).
As a field, surface diffusion has evolved
somewhat separately from solid-state dif- fusion, perhaps because of the very differ- ent techniques employed. This separate- ness has resulted in some inconsistencies in nomenclature between the two fields. It is appropriate here to discuss these briefly. For both self- and hetero-diffusion it is usual to refer to the motion of atoms for short distances, where there is one type of site, as “intrinsic diffusion”. When the con- centration of diffusion species is very low the relevant diffusion coefficient is called a “tracer” diffusion coefficient. This is not
in fact the diffusion coefficient obtained with radioactive tracers but a single parti- cle diffusion coefficient. (A single particle can be followed or traced.) At higher con- centrations, the relevant diffusion coeffi- cient is called a chemical diffusion coeffi- cient. Thus “intrinsic” diffusion is formally the same as diffusion in the pseudo-one- component system, as described in Sec. 3.2.2.3.
When the surface motion of atoms ex-
tends over long distances, and many types of site are encountered (this is macroscopic diffusion in contrast with the microscopic or “intrinsic” diffusion discussed above) the relevant diffusion coefficient is called a mass transfer diffusion coefficient. Fur- ther discussion of these and relationships among these diffusion coefficients are re- viewed by Bonzel (1990).
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233. 3.2 Macroscopic Diffusion 181
3.2.3 Phenomenological Equations
of Irreversible Thermodynamics
An implication of Fick’s First Law (Eq.
(3-1)) is that once the concentration gradi-
ent for species ireaches zero, all net flow
for species istops. Although frequently
correct, this is rather too restrictive as a
condition for equilibrium. In general, net
flow for species ican cease only when all
direct or indirect forces on species iare
zero. This is conveniently handled by pos-
tulating linear relations between each flux
and all the driving forces. We have for flux
iin a system with kcomponents
(3-22)
where the L s are called phenomenological
coefficients and X
iis the driving force on
component iand is written as – grad
m
i,
where
m
iis the chemical potential of spe-
cies i. X
ican also result from an external
driving force such as an electric field. In
this case X
i=Z
ieE, where Z
iis the number
of changes on i, eis the electronic charge,
and Eis the electric field. For ionic con-
ductors Z
iis the actual ionic valence. For
alloys Z
iis an “effective” valence which is
usually designated by the symbol Z
i
*. Z
i
*
consists of two parts, the first, Z
i
el, repre-
sents the direct electrostatic force on the
moving ion (Z
i
elis expected to be the ion’s
nominal valence), and the second, Z
i
wd,
accounts for the momentum transfer be-
tween the electron current and the diffusing
atom. A comprehensive review of all as-
pects of Z
i
*and the area of electromigra-
tion has been given by Huntington (1975).
X
qis a driving force resulting from a tem-
perature gradient (if present). X
qis given
by –T
–1
gradT. When referring to diffu-
sion in a temperature gradient it is usual to
let L
iqbe expressed as
LQL
iq
k
n
kik=
=1
∑
*
JLXLX
i
k
ik k iq q=∑ +
where Q
k
*is called the “heat of transport”
for species k(Manning, 1968). This can
be further related to the heat of transport derived from the actualheat flow. This
subject is dealt with further by Manning (1968) and Philibert (1991). Sometimes it is convenient to discuss the flux in an electric field occurring simultaneously with diffusion in a chemical potential gradient. A simple example would be diffusion from a tracer source in an electric field – the so- called Chemla experiment (Chemla, 1956). In such a case, we write X
ias
X
i= – gradm
i+ Z
ieE (3-23)
Now the phenomenological coefficient
L
iq(Eq. (3-22)) describes the phenomenon
of thermal diffusion, i.e., the atom flow re- sulting from the action of the X
q, i.e., the
temperature gradient (Soret effect). There is also an equation analogous to Eq. (3-22) for the heat flow itself:
(3-24)
L
qkis a heat flow phenomenological coef-
ficient, i.e., the heat flow which accompa- nies an atom flow (the Dufour effect). L
qq
refers directly to the thermal conductivity.
The other phenomenological coeffi-
cients, L
ikin Eq. (3-22), are concerned
with the atomic transport process itself. The off-diagonal coefficients are con- cerned with “interference” between the at- oms of different types. This may arise from interactions between A and B atoms and/or the competition of A and B atoms for the defect responsible for diffusion, see Sec. 3.3.1.8. A most important property of the phenomenological coefficients is that they are independent of driving force. Further,
the matrix of Lcoefficients is symmetric:
this is sometimes called Onsager’s theorem or the reciprocity condition. That is,
L
ij= L
ji (3-25)
JLXLX
q
k
qk k qq q=∑ +www.iran-mavad.com
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234. The set of equations represented by Eq.
(3-22) are normally called the “phenomen-
ological equations” or Onsager equations
and were postulated as a central part of the
theory of irreversible processes. Details of
the theory as it applies to diffusion can be
found in many places but see especially the
reviews by Howard and Lidiard (1964),
Adda and Philibert (1966), and Allnatt and
Lidiard (1993).
The phenomenological coefficients are
sometimes said to have a “wider meaning”
than quantities such as the diffusion coeffi-
cients or the ionic conductivity. The wider
meaning comes about because the pheno-
menological coefficients do not depend on
driving force but only on temperature and
composition. In principle, armed with a full
knowledge of the Ls, the technologist
would have the power, if not to control the
diffusive behavior of the material, at least
to predict the diffusive behavior no matter
what thermodynamic force or forces or
combination thereof were acting.
Unfortunately, the experimental determi-
nation of the Ls is most difficult in the solid
state. This is in contrast, incidentally, to
liquids, where, by use of selectively perme-
able membranes, measurement of the Ls is
possible. The reader might well ask then
why the L s were introduced in the first
place when they are essentially not amen-
able to measurement! There are several
reasons for this. First, relations can be de-
rived between the Ls and the (measurable)
tracer diffusion coefficients – we will dis-
cuss this further in Sec. 3.2.3.4. Further,
the Onsager equations provide a kind of ac-
counting formalism wherein an analysis
using this formalism ensures that once the
components and driving forces have all
been identified nothing is overlooked, and
that the whole is consistent; this will be
discussed in Secs. 3.2.3.1 and 3.2.3.2.
Finally, many “correlations” in diffusion,
e.g., the tracer correlation factor and the
vacancy-wind effect, can be conveniently
expressed in terms of the Ls. This can
sometimes aid in understanding the nature
of these complex correlations. A most im-
portant contribution to this was made by
Allnatt (1982), who related the Ls directly
to the microscopic behavior – see Sec.
3.3.1.8.
In the next sections we shall restrict our-
selves to isothermal diffusion and focus on
the relations which can be derived among
the Ls and various experimentally acces-
sible transport quantities.
3.2.3.1 Tracer Diffusion
Let us first consider tracer diffusion in a
pure crystal. The general strategy is (1) to
describe the flux of the tracer with the On-
sager equations (Eq. (3-22)) and (2) to de-
scribe the flux with Fick’s First Law (Eq.
(3-1)) and then equate the two fluxes to
find expressions between the diffusion co-
efficient and the phenomenological coeffi-
cients. We shall deal with this in detail to
show the typical procedure (see, for exam-
ple, Le Claire (1975)). We can identify two
components, A and its tracer A*. We write
for the fluxes of the host and tracer atoms,
respectively,
J
A= L
AAX
A+ L
AA*X
A* (3-26)
and
J
A*= L
A*A*X
A*+ L
A*AX
A (3-27)
Strictly, the vacancies should enter as a
component but it is unnecessary here be-
cause of the lack of a vacancy gradient.
Since the tracer atoms and host atoms
mix ideally (they are chemically identical),
it can easily be shown that the driving
forces are
(3-28)
X
kT
c
c
x
A
A
A=−
∂
∂
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235. 3.2 Macroscopic Diffusion 183
and
(3-29)
where c
iis the mole fraction. This leads
immediately to
(3-30)
It is convenient to introduce the quantity
C
i(=Nc
i/V) which is the number of atoms
of type iper unit volume and Nis the num-
ber of entities (A + A*), and Vis the vol-
ume. Since there must be a symmetry con-
dition ∂c
A/∂x=–∂c
A*/∂x, we now find that
(3-31)
However, Fick’s First Law, Eq. (3-1), states
that
(3-32)
and so the tracer diffusion coefficient is
given by
(3-33)
When c
A*Æ0, which is the usual situation
when tracers are used experimentally, we
find that
D
A*= kTVL
A*A*/c
A*Nc
A*Æ0 (3-34)
Since the fluxes J
Aand J
A*are always
equal but opposite in sign, then going
through the above procedure, but now for
J
A, leads to the relation
(3-35)
Now let us relate the diffusion coefficient
to the ionic conductivity in order to obtain
LL
c
LL
c
AA A*A
A
AA* A*A*
A*
=
++
D
kTV
N
L
c
L
c
A*
A* A*
A*
A* A
A= −
⎛
⎝
⎜
⎞
⎠
⎟
JD
C
x
A* A*
A*=−
∂
∂
J
C
x
kTV
N
L
c
L
c
A*
A* A* A
A
A* A*
A*=
∂
∂
−
⎛
⎝
⎜
⎞
⎠
⎟
JL
kT
c
c
x
L
kT
c
c
x
A* A* A*
A*
A*
A* A
A
A=−
∂
∂
−
∂
∂
X
kT
c
c
x
A*
A*
A*=−
∂
∂
an expression for the Haven ratio H
R(see
also Sec. 3.3.2).
We consider the system of A and A* in
an electric field. We will assume that the
A and A* are already mixed. We will now
let X
A=X
A*=ZeE, where Zis the number
of charges carried by each atom. From Eq.
(3-27) the flux of A* is
J
A*= Z
A*eE(L
A*A*+ L
A*A) (3-36)
The drift mobility u
A*is related to the flux
by
J
A*= C
A*u
A*E (3-37)
and so
(3-38)
The mobility u
Ais equal to u
A*since A and
A* are chemically identical.
By means of the so-called Nernst–Ein-
stein equation (Eq. (3-130)), the mobility
can be converted to a dimensionally correct
diffusivity, D
s, i.e.,
(3-39)
As is discussed in detail in Sec. 3.3.2, D
s
does not have a meaning in the sense of
Fick’s First Law (Eq. (3-1)). Its meaning is
the diffusion coefficient of the assemblyof
ions as if the assembly itself acts like a sin-
gle (hypothetical) particle. It is conven-
tionally related to the tracer diffusion coef-
ficient by the Haven ratio H
Rwhich is de-
fined as:
(3-40)
Using the equation for D
A*we find, after
letting c
A*Æ0, that
(3-41)
H
L
LL
R
A* A*
A* A* A* A=
+
H
D
D
R
A*≡
s
D
kTV
Nc
LL
s=
A*
A* A* A* A
()+
u
ZeV
N
LL
c
A*
A* A* A* A* A
A*=
+⎛
⎝
⎜
⎞
⎠
⎟www.iran-mavad.com
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236. For a pure crystal and where the vacancy
concentration is very low, H
Rcan be iden-
tified directly with the tracer correlation
factor f(see Sec. 3.3.2) and
(3-42)
Note that if L
A*A= 0, i.e., there is no inter-
ference between tracer and host, then
H
R=1. More generally, when the defect
concentration is high and there are corre-
lations in the ionic conductivity (see Sec.
3.3.1.7), we find that
(3-43)
and f
Iis the physical correlation factor.
3.2.3.2 Chemical Diffusion
For chemical diffusion between A and B
when, say, the vacancy mechanism is oper-
ating (see Sec. 3.3.1.1), the Onsager equa-
tions are written as (Howard and Lidiard,
1964)
J
A= L
AAX
A+ L
ABX
B (3-44)
J
B= L
BBX
B+ L
BAX
A (3-45)
Strictly, the vacancies should enter here as
another species, so that we would write
J
A= L
AAX
A+ L
ABX
B+ L
AVX
V (3-46)
J
B= L
BBX
B+ L
BAX
A+ L
BVX
V (3-47)
and, because of conservation of lattice
sites,
J
V= – (J
A+J
B) (3-48)
The usual assumption is that the vacancies
are always at equilibrium and X
V=0. For
this to happen the sources and sinks for va-
cancies, i.e., the free surface, dislocations,
or grain boundaries, must be effective and
sufficiently numerous.
Hff
L
LL
RI
A* A*
A* A* A* A==/
+
Hf
L
LL
R
A* A*
A* A* A* A==
+
Following the same kind of procedure as
before (see Sec. 3.2.3.1), but recognizing that generally A and B do notideally mix,
we find that rather than by Eq. (3-28), X
iis
now given by
(3-49)
where
gis the activity coefficient of either A
or B. We find that the diffusion coefficient of, say A, actually an intrinsic diffusion co-
efficient (see Sec. 3.2.2.4), is given by
(3-50)
If we want to find a relation between D
I
A
and the tracer diffusion coefficient, say D
A*, we would need to introduce A* for-
mally into the phenomenological equations which would now become (Stark, 1976; Howard and Lidiard, 1964)
J
A= L
AAX
A+ L
AA*X
A*+ L
ABX
B(3-51)
J
A*= L
A*A*X
A*+ L
A*AX
A+ L
A*BX
B(3-52)
J
B= L
BBX
B+ L
BAX
A+ L
BA*X
A*(3-53)
These equations can be developed to give
(3-54)
and
(3-55)
When these two equations are combined
we find that
(3-56)
DD
c
c
Lc
L
c
L
c
A
I
A*
A*
A* A* A
A* A
A*
AB
B=1
1
+
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
×+ −
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
ln
ln
g
D
kTV
N
LL
c
LL
c
c
A
I AA A*A
A
AB A*B
B=
+
−
+⎛
⎝
⎜
⎞
⎠
⎟
×+
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
ln
ln
1
g
D
kTV
N
L
c
L
c
A*
A* A*
A*
A* A
A= −
⎛
⎝
⎜
⎞
⎠
⎟
D
kTV
N
L
c
L
cc
A
I AA
A
AB
B= −
⎛
⎝
⎜
⎞
⎠
⎟+
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟1
ln
ln
g
X
kT
c
c
xc
i
i
i=−
∂
∂
+
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟1
ln
ln
g
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237. 3.2 Macroscopic Diffusion 185
This is in fact the so-called Darken equa-
tion (Darken, 1948), relating the intrinsic
diffusion coefficient to the tracer diffusion
coefficient but with the addition of the term
in brackets. This term is sometimes called
the “vacancy-wind-term” and does not
vary much from unity. Note that it results
from non-zero cross terms L
A*Aand L
AB.
The approach from the phenomenological
equations has told us only of the formal ex-
istence of this term, but an evaluation of it
requires consideration of the detailed mi-
croscopic processes which generate the
correlations contained in the cross terms.
This is considered in Sec. 3.3.1.8.
These examples suffice to show how the
phenomenological equations are useful in
presenting a consistent and unified picture
of the diffusion process, no matter how com-
plex. For further information on the subject,
the reader is directed to the classic review by
Howard and Lidiard (1964) and also many
other treatises, such as those by Adda and
Philibert (1966), Le Claire (1975), Stark
(1976), Kirkaldy and Young (1987), Phili-
bert (1991), and Allnatt and Lidiard (1993).
3.2.3.3 Einsteinian Expressions for
the Phenomenological Coefficients
An important development in the area of
solid-state diffusion was the fact that the
phenomenological coefficients can be ex-
pressed directly in terms of atomisticEin-
steinian formulae (Allnatt, 1982).
(3-57)
where DR
i
(t) is the total displacement of
species iin time t, Vis the volume, and the
Dirac brackets denote a thermal average A.
It is important to note that DR
(i)
is the sum
of the displacements of the individual par-
ticles of type i. In effect, DR
(i)
is the dis-
LV kTt
tt
ij
Vt
ij= lim lim ( )
() ()
() ( )
→∞ →∞
−
×〈 ⋅ 〉
6
1
DDRR
placement of the system of species ias if
that system were a particle itself. The diag- onal phenomenological coefficients derive from correlations of the system of species i
with itself, i.e., self-correlations. This does notmean tracer correlations here: they are
correlations in the random walk of an atom. The correlations here are in the random walk of the system“particle”. The off-diag-
onal terms derive from interference of the system of species iwith the system of spe-
cies j. In effect it is mathematically equiva-
lent to two systems iand jbeing treated
like two interfering “particles”.
Eq. (3-57) has been very useful for cal-
culating the L
ijby means of computer sim-
ulation – see the pioneering calculations by Allnatt and Allnatt (1984). Much of that material has been reviewed by Murch and Dyre (1989). A brief discussion is given in Sec. 3.3.1.8.
3.2.3.4 Relating Phenomenological
Coefficients to Tracer Diffusion
Coefficients
For the case of multi-component alloys,
Manning (1968, 1970, 1971) derived expres-
sions relating the phenomenological coeffi-
cients to tracer diffusion coefficients. The ex-
pressions were developed for a particular
model for concentrated alloys, the so-called
random alloy model. In this model the atomic
components are randomly distributed over
the available sites, the vacancy mechanism is
assumed and the exchange frequencies of the
atomic components with the vacancies de-
pend only on the nature of the atomic compo-
nents and not on their environment.
For a binary system the relations are
L
N
kT
cD
f
f
cD
cD cD
AA A A
AA
AA BB=( 3-58
)*
*
**
()
()
×+
−
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
1
1
0
0www.iran-mavad.com
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238. (3-59)
where f
0is the tracer correlation factor in
the lattice of either pure component, see
Sec. 3.3.1.3. The relation for L
BBis ob-
tained from L
AAby interchanging B with A.
Lidiard (1986) showed that Eqs. (3-58)
and (3-59) can in fact be obtained without
recourse to the random alloy model, on the
basis of two microscopic assumptions
which, although intuitive in nature, are in-
dependent of any microscopic model ex-
cept for the inclusion of the tracer correla-
tion factor in the pure lattice f
0, which is
dependent on mechanism and structure.
Lidiard’s findings immediately suggested
that Eqs. (3-58) and (3-59) are probably ap-
propriate for a much wider range of alloys
than are reasonably represented by the ran-
dom alloy model itself. Bocquet (1987)
also found relations of the same form for
the random alloy when interstitial mecha-
nisms are operating. With Monte Carlo
computer simulation, Zhang et al. (1989a)
and Allnatt and Allnatt (1991) explored the
validity of Eqs. (3-58) and (3-59) in the
context of the interacting binary alloy
model described in Sec. 3.3.1.5. It was
found that Eqs. (3-58) and (3-59) apply
very well except, perhaps surprisingly, at
compositions approaching impurity levels.
The breakdown there is readily traced to
the fact that these equations have built into
them the requirement that
w
4=w
0(see Sec.
3.3.1.4 for the impurity frequency nota-
tion). For many cases
w
3also equals w
0
and so this is equivalent to a condition of
no vacancy-impurity binding. This is not
surprising given that the equations origi-
nated with the random alloy model where
this requirement is automatically fulfilled.
Eqs. (3-58) and (3-59) have also been
derived for B1 and B2 ordered alloys (Be-
lova and Murch, 1997).
L
N
kT
f
f
cDcD
cD cD
AB
AABB
AA BB=
**
**()
()
1
0
0−
+
A most important consequence of Eqs.
(3-58) and (3-59) is the Darken–Manning expression which relates the interdiffusion coefficient D˜, the tracer diffusion coeffi-
cients D
A*and D
B*and the thermodynamic
factor (1 +∂ln
g/∂c):
(3-60)
where the vacancy wind factor Sis given by
(3-60a)
Eq. (3-60) is not only appropriate for
random alloys, but also for ordered alloys/ intermetallic compounds with B1 and B2 structures. Recent computer simulations have also shown that Eq. (3-60) is a good approximation for ordered alloys/intermet- allic compounds with L1
2, D0
3, and A 15
structures (Murch and Belova, 1998).
Further discussion on relations between
phenomenological coefficients and tracer diffusion coefficients, and impurity diffu- sion coefficients, can be found in the re- views by Howard and Lidiard (1964), Le Claire (1975), and Allnatt and Lidiard (1987, 1993).
3.2.4 Short-Circuit Diffusion
It is generally recognized that the rate at
which atoms migrate along grain boundar-
ies and dislocations is higher than that
through the lattice. From an atomistic point
of view it is very difficult to discuss, in a
precise way, the diffusion events in such
complex and variable situations. Much of
the understanding has been gained from
computer simulations using the Molecular
Dynamics and Lattice Statics methods;
see, for example, Kwok et al. (1984) and
Mishin (1997). In this section we shall dis-
S
fccD D
fcD cD cD cD
=
AB A* B*
A A* B B* B A* A B*
1
1
0
2
0
+
−−
++
() ( )
()()
˜
()
ln
DcD cD
c
S=
BA* AB*++
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟1
g
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239. 3.2 Macroscopic Diffusion 187
cuss short-circuit diffusion from a pheno-
menological point of view.
Fisher (1951) was probably the first to
introduce a distinct diffusion coefficient
for material migrating along the short-cir-
cuit path. For grain boundaries, each one is
conceived as a slab of width
dfor which
the averagediffusion coefficient is D¢. For
dislocations, each one is conceived as a
pipe of diameter 2afor which the average
diffusion coefficient is D ¢.
For conventional diffusion experiments
it has been usual to distinguish three dis-
tinct regimes, depending on the magnitude
of the lattice diffusion length ÷
---
D
lt, where
D
lis the lattice diffusion coefficient and tis
time (Harrison, 1961).
Regime A: ÷
---
D
ltis much larger than the
spacing lbetween the short-circuit paths.
For grain boundaries lis the average diam-
eter of the grains and for dislocations lis
the distance between dislocation pinning
points. Diffusion from adjacent short cir-
cuits overlap extensively. This condition
is met for small-grained materials or very
long diffusion times. Hart (1975) proposed
that an effective diffusion coefficient D
eff
can be introduced which still satisfies solu-
tions to Fick’s Second Law and can be
written as
D
eff= ≤≤D¢+ (1 –≤≤)D
l (3-61)
where ≤≤is the fraction of all sites which
belong to the short-circuit paths. Eq. (3-61)
is usually called the Hart Equation. Le
Claire (1975) has given a rough calculation
using Eq. (3-61). For typical dislocation
densities in metals, ≤≤=10
–8
. The disloca-
tion contributionto a measured or effective
Dwill then exceed 1% when D¢/D>10
6
.
Because the activation energy for lattice
diffusion in metals is about 34T
M.Pt.,
where T
M.Pt.is the melting temperature
and the activation energy for short-circuit
diffusion is generally about half that of
volume diffusion, then this condition
D¢/D>10
6
is usual for temperatures below
about half the melting temperature. This is
why experiments where lattice diffusion
only is of interest tend to be made at tem-
peratures aboveabout half the melting tem-
perature.
Impurities are often bound to short-cir-
cuit paths, in which case the Hart Equation
(Eq. (3-60)) is written as
D¢= ≤≤D¢exp (E
b/kT) + (1 –≤≤)D
l(3-62)
where E
bis the binding energy of the im-
purity and the diffusion coefficients refer
to impurities. The Hart Equation can be
shown to be fairly accurately followed
when l/÷
---
D
lt≤0.3 and also for l/÷
---
D
lt≥100
(this corresponds to Regime C – see the
next paragraph) (Murch and Rothman,
1985; Gupta et al., 1978).
Regime C: It is convenient to discuss the
other limit out of sequence. When ÷
---
D
ltis
much smaller than the distances lbetween
the short-circuit paths, we have Regime C
kinetics. In this instance, all material
comes down the short-circuit paths and the
measured diffusion coefficient is given en-
tirely by D¢.
Regime B: In this intermediate case, it is
assumed that ÷
---
D
ltis comparable to lso
that material which is transported down a
short-circuit path and which diffuses out
into the lattice is unlikely to reach another
short-circuit path.
There are various solutions of Fick’s
Second Law available to cope with tracer
diffusion in the presence of short-circuit
paths, but space prevents us from dealing
with these in any detail: they are discussed
at length by Kaur, Mishin and Gust (1995).
Suzuoka (1961) and Le Claire (1963) have
given a solution for the grain boundary
problem for the usual case when there is a
finite amount of tracer originally at the sur-
face, see Eq. (3-8). It is found, among otherwww.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

240. things, that
(3-63)
Hence the product D¢
dcan be found by de-
termining the slope of the linearregion
(penetrations reached by grain boundary
diffusion) in a plot of lnCversus x
6/5
(not
the usual x
2
) and also with a knowledge of
D
litself. It can be difficult, however, to ob-
tain the two diffusion coefficients in one
experiment; the practicalities of this and
alternatives are discussed by Rothman
(1984). An example of a tracer penetration
plot with a clear contribution from grain
boundary diffusion (x
6/5
dependence) is
shown in Fig. 3-3.
Le Claire and Rabinovitch (1984) have
addressed the dislocation pipe problem
d
d
=
ln
().
/
//
C
x
D
t
D
l
65
53 12
14
066
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
′
−
d
and provided near-exact solutions to Fick’s
Second Law for both an isolated disloca-
tion pipe and arrays of dislocation pipes.
The profiles generally are not unlike the
grain boundary ones except that a linear re-
gion in a plot of lnCversus xis now found,
i.e.,
(3-64)
where Ais a slowly varying function of
a/(Dt)
1/2
. An example of such a tracer pen-
etration plot with a clear contribution from
dislocation pipe diffusion (xdependence)
is shown in Fig. 3-4.
For further details on grain boundary
diffusion we refer to reviews by Peterson
(1983), Mohan Rao and Ranganathan
(1984), Balluffi (1984), Philibert (1991),
d
d
=
ln ( )
[( / ) ]
/
C
x
Aa
DD a
l
−
′−1
212
188 3 Diffusion Kinetics in Solids
Figure 3-3.Tracer concentration
profiles of
203
Pb into polycrystal-
line Pb showing a contribution
from grain boundaries (Gupta and
Oberschmidt, 1984).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

241. 3.3 Microscopic Diffusion 189
Kaur et al. (1995), Mishin et al. (1997). For
dislocation pipe diffusion we refer to Le
Claire and Rabinovitch (1984) and Phili-
bert (1991).
3.3 Microscopic Diffusion
3.3.1 Random Walk Theory
We have already seen in Sec. 3.2 that the
phenomenological or continuum theory of
diffusion is set up in terms of quantities
such as the diffusion coefficients, the ionic
conductivities, or more generally, the phen-
omenological coefficients, L
ij. For the ma-
terials technologist this theory provides a
perfectly suitable formalism to describe the
macroscopic features arising from the dif-
fusion of atoms in solids. By its very na-
ture, this macroscopic theory makes no ref-
erence to the discrete atomic events which
give rise to macroscopically observable
diffusion. To describe the atomic events, a
wholly separate theory, termed “Random
Walk Theory” has been formulated. This
theory is based on the premise that macro-
scopic diffusion is the net result of many
individual atomic jumps. The theory at-
tempts to relate the quantities such as the
diffusion coefficients, the ionic conductiv-
ities, or, more generally, the L
ij, in terms of
lattice and atomic characteristics, notably
jump frequencies. Although originally
quite precise in use (Howard, 1966), the
term “Random Walk Theory” is now used
rather loosely to describe any mathematical
approach that focuses on the sequence of
jumps of atoms in the solid state. Inextri-
cably linked with this theory is the exis-
tence of correlated random walks of atoms,
in other words, walks where there is an ap-
parent memory between jump directions.
Much of random walk theory is concerned
with describing such correlations.
3.3.1.1 Mechanisms of Diffusion
As a prelude to a discussion of random
walk theory, in this section we will briefly
discuss the common mechanisms of solid-
state diffusion.
Interstitial Mechanism
In the interstitial mechanism, see Fig.
3-5, sometimes called the direct intersti-
tial mechanism, the atoms jump from one
interstice to another without directly in-
volving the remainder of the lattice. Since
the interstitial atom does not need to “wait”
to be neighboring to a defect in order to
jump (in a sense it is always next to a va-
cancy), diffusion coefficients for atoms mi-
grating with this mechanism tend to be
fairly high. As would be expected, atoms
Figure 3-4.Tracer concentration profiles of
22
Na
into single crystal NaCl showing a contribution from
dislocations (Ho, 1982).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

242. such as H, N, O, and C diffuse in this way
in metals. It should also be noted that with-
out the defect being required to affect the
jump, no defect concentration term and
therefore defect formation energy enters
the activation energy for diffusion; see Sec.
3.3.4.
Interstitialcy Mechanism
In the interstitialcy mechanism, see Fig.
3-6, sometimes called the indirect intersti-
tial mechanism, two atoms, one an intersti-
tial and the other an atom on a regular lat-
tice site, move in unison. The interstitial
atom moves to a regular site, whereas the
regular site atom moves to an interstice.
Collinear and non-collinear versions are
possible depending on constraints imposed
by the lattice.
Because the interstitial formation energy
is generally very high in metals, the equi-
librium concentration of interstitials is very
small and their contribution to diffusion is
unimportant in most cases. However, for
plastically deformed or irradiated metals
the concentration of interstitials (besides
vacancies) can be appreciable. It should be
noted that the interstitial thus formed is not
located on an interstice, but in a dumbbell
split configuration. The migration possibil-
ities of dumbbells have been discussed by
Schilling (1978).
As a result of measurements of the
Haven ratio (see Sec. 3.3.2), the interstis-
tialcy mechanism appears to be highly
likely for silver diffusion in AgBr (Friauf,
1957). Note that with the interstitialcy
mechanism, say the collinear version, a
tracer atom moves a distance rwhereas the
charge apparently moves a distance 2r.
This needs to be taken into account in the
interpretation of the Haven ratio. The inter-
stitialcy mechanism probably occurs rela-
tively frequently in ionic materials, espe-
cially those with open lattices such as the
anion lattice in the fluorite structure or
those which are highly defective. Nonethe-
less, there is often so much attendant local
relaxation around the interstitial that far
more complex quasi-interstitialcy mecha-
nisms could well operate. An example is
oxygen diffusion in UO
2+x, where the ex-
cess oxygen is located as di-interstitials
with relaxation of two oxygen atoms from
regular sites to form two new interstitials
and two new vacancies. The actual mecha-
nism of oxygen transport is not known
but a quasi-interstitialcy mechanism is un-
doubtedly responsible (Murch and Catlow,
1987).
We should also mention that the interstit-
ialcy mechanism appears to be very impor-
tant in self-diffusion in silicon and possibly
certain substitutional solutes in silicon are
also transported via this mechanism (Frank
et al., 1984).
190 3 Diffusion Kinetics in Solids
Figure 3-5.Interstitial diffusion mechanism.
Figure 3-6.Interstitialcy diffusion mechanism.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

243. 3.3 Microscopic Diffusion 191
Vacancy Mechanism
The most important diffusion mecha-
nism of all is undoubtedly the vacancy
mechanism, shown in Fig. 3-7. A substitu-
tional solute atom or an atom of the solvent
diffuses simply by exchanging places with
the vacancy. There can be attractive or re-
pulsive interactions of the solute with the
vacancy, which can have a profound in-
fluence on the diffusion coefficient of the
solute and to a lesser extent the solvent
by way of correlation effects. This is dis-
cussed in Secs. 3.3.1.4 and 3.4.2. The va-
cancy mechanism is definitely the pre-
ferred mechanism for metals and alloys for
both host and substitutional solutes. In
most other materials the vacancy appears
to play the major diffusion role except
when the concentration of interstitials pro-
duced by nonstoichiometry or doping (in
oxides) or irradiation (in metals) is so high
that contributions from interstitialcy or
similar mechanisms become dominant.
Aggregates of vacancies such as the
divacancy or trivacancy can also contribute
to diffusion. These appear to be important
at high temperatures in metals where their
contribution has been largely inferred from
curvatures of the Arrhenius plot (logDvs.
1/T); see, for example, the f.c.c. metals
(Peterson, 1978) (see Sec. 3.4.1.1). Bound
vacancy pairs, i.e., a cation vacancy bound
to an anion vacancy, are important contrib-
utors to tracer diffusion in alkali halides
(Bénière et al., 1976).
Interstitial-Substitutional Mechanism
On occasion, solute atoms may dissolve
interstitially and substitutionally. These
solute atoms may diffuse by way of the
dissociative mechanism(Frank and Turn-
bull, 1956) and/or the kickout mechanism
(Gösele et al., 1980). In bothmechanisms
the interstitial solute diffuses rapidly by the
interstitial mechanism. In the dissociative
mechanism the interstitial combines with a
vacancy to form a substitutional solute. At
a later time this substitutional can dissoci-
ate to form a vacancy and an interstitial so-
lute (really a Frenkel defect). The anoma-
lously fast diffusion of certain solutes, e.g.,
Cu, Ag, Au, Ni, Zr, and Pd in Pb, appears
to have a contribution from the dissocia-
tive mechanism (Warburton and Turnbull,
1975; Bocquet et al., 1996). In the kickout
mechanism, on the other hand, the solute
interstitial uses the interstitialcy mecha-
nism to involve the regular site lattice. In
the process a host interstitial is formed and
the interstitial solute then occupies a sub-
stitutional site. This process can be re-
versed at a later stage. The kickout mecha-
nism appears to operate for rapid diffusion
of certain foreign atoms, such as Au in Si
(Frank et al., 1984).
Exchange Mechanism
The exchange mechanism, in which two
neighboring atoms exchange places, has in
the past been postulated as a possible diffu-
sion mechanism. The existence of the Kir-
kendall effect in many alloy interdiffusion
experiments (this implies that the respec-
tive intrinsic diffusion coefficients are un-
equal, which is not possible with the ex-
change mechanism, see Sec. 3.2.2.4) and
the very high theoretical activation ener-
Figure 3-7.Vacancy mechanism.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

244. gies in close-packed solids suggest that this
mechanism is unlikely. It may, however,
occur in liquids and in quenched-liquids
such as amorphous alloys (Jain and Gupta,
1994). Ring versions of the exchange
mechanism certainly have lower theoreti-
cal activation energies but require substan-
tial cooperation among the atoms, which
seems unlikely.
Surface Diffusion Mechanisms
A number of mechanisms for surface
diffusion on metals (by far the most studied
class of material) have been postulated.
They include activated hopping of ad-
sorbed atoms from one surface site to an-
other, where the jump distance is simply
the distance between sites. Similarly, a va-
cancy ina terrace, i.e., surface vacancy,
can also move in the same sort of way
within the terrace. At low temperatures and
rough surfaces, exchange between an ad-
sorbed atom and an atom in the substrate is
predominant. At high temperatures non-lo-
calized diffusion and surface melting are
also possible. The mechanisms for surface
diffusion are discussed further in the book
edited by Vu Thien Binh (1983) and in the
reviews by Rhead (1989) and Bonzel
(1990).
3.3.1.2 The Einstein Equation
Let us consider atoms (of one chemical
type) diffusing in their concentration gradi-
ent in the xdirection. It can readily be
shown (see, for example, Adda and Phili-
bert, 1966; Manning, 1968; Le Claire,
1975) that the netflux of atoms across a
given plane x
0is given by
(3-65)
JCx
t
C
xt
Cx
C
xC t
x=()
()
0
2
0
2
2
2
〈〉
−
∂
∂
〈〉
−
∂
∂
∂
∂
〈〉
XX
X
plus higher order terms. In Eq. (3-65), C(x
0)
is the concentration of the diffusing atoms at x
0, ·XÒis the mean displacement or drift,
and ·X
2
Òis the mean squared displacement.
The Dirac brackets ·Òdenote an average
over all possible paths taken in time t.
In a situation where diffusion properties
do notdepend on position, e.g., diffusion
of tracer atoms in a chemicallyhomogene-
ous system, the third term in Eq. (3-65) is zero. The term containing the drift ·XÒis
also zero because in a chemically homoge- neous system the probability that, say, a tracer atom migrates some distance +Xin
time t(starting from x
0) equals the prob-
ability that an atom migrates a distance –X.
Similarly, other odd moments are zero. Provided that other higher-order terms can be neglected, Eq. (3-65) reduces to
(3-66)
By comparison with Fick’s First Law, Eq. (3-1), Eq. (3-66) immediately gives
(3-67)
Eq. (3-67) is called the Einstein Equation, probably the single most important equa- tion in the theory of diffusion kinetics. The superscript * indicates that the diffusion coefficient refers to tracer atom diffusion in a chemically homogeneous system. Equations for D
y* and D
z* have the same
form as Eq. (3-67). For three-dimensional isotropic crystals the tracer diffusion coef- ficient is the same in every direction and
(3-68)
where Ris the vector displacement of an
atom in time t . For two-dimensional situa-
tions the factor 6 in Eq. (3-68) is replaced by 4.
D
t
*=
〈〉R
2
6
D
t
x
*=
〈〉X
2
2
J
C
xt
x=
∂
∂
〈〉X
2
2
192 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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245. 3.3 Microscopic Diffusion 193
3.3.1.3 Tracer Correlation Factor
Either Eq. (3-67) or (3-68) can provide
the starting point for a discussion of corre-
lation effects in the diffusion walk. For
convenience we shall use Eq. (3-68). Let us
consider the atomistic meaning of the dis-
placement R. It is simply the sum of nsuc-
cessive jump vectors, r
1, r
2, r
3,…,r
n
(3-69)
The squared displacement R
2
is then simply
(3-70)
The average of the squared displacement
·R
2
Òequals of course the sum of the aver-
ages and we have
(3-71)
For a complete random walk, i.e., where
each direction is unrelated to the previous
one, the second term in Eq. (3-71) is zero
because for any product r
i·r
i+jthere will
always be another of opposite sign. We
now have
(3-72)
Because in most cases atoms require the
assistance of point defects in order to move
about (see Sec. 3.3.1.1), there is generally
an unavoidable memory or correlation ef-
fect between jump directions. In order to
appreciate this, let us focus on Fig. 3-8a
where the vacancy mechanism is shown. In
this figure one atom symbol is shown filled
to indicate that it is a tracer and can be fol-
lowed. Let us assume that the tracer and the
vacancy have just exchanged places. Be-
cause the vacancy is still neighboring to the
tracer the next jump of the traceris quite
likely to cancel out the previous jump. The〈〉 〈〉∑Rr
2
1
2
=
=i
n
i
〈〉 〈〉+ 〈⋅ 〉∑∑∑
−−
+
Rr rr
2
1
2
1
1
1
2=
===i
n
i
i
n
j
ni
iij
Rr rr
2
1
2
1
1
1
2=
===i
n
i
i
n
j
ni
iij
∑∑∑+⋅
−−
+
Rr=
=i
n
i
1
∑
probability of doing this is in fact exactly 2/gwhere gis the lattice coordination
(Kelly and Sholl, 1987). On the other hand, the tracer has a reducedprobability of con-
tinuing in the direction of the first jump, since this requires the vacancy to migrate to point A. In other words, there is a mem- ory or correlation between directions of tracer jumps. In no way does this imply that the vacancy somehow favors the tracer atom. In fact, in this example atoms sur- rounding the vacancy jump randomly with the vacancy so that the vacancy itself moves on an uncorrelated random walk with no memory or correlation whatsoever between its jump directions. It should be noted that weaker correlations in the tracer jumps also come about as long as the va-
Figure 3-8.(a) Correlation effects arising from the
vacancy mechanism – see text. (b) Correlation ef-
fects arising from the interstitialcy mechanism (see
text).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

246. cancy remains in the vicinity of the tracer
and can approach the tracer from a non-
random direction.
Let us consider Fig. 3-8b, where the col-
linear interstitialcy mechanism is assumed.
Let us assume that the tracer is in the inter-
stitial position. The first jump (a pair of
atoms move) will take the tracer immedi-
ately to a regular lattice site and a new
atom, formerly at a regular lattice site,
takes up a position as the interstitial. The
pair of atoms which next moves possibly
again involves the tracer atom and in doing
so will cancel out the previous tracer jump.
Like the vacancy mechanism, the memory
or correlation between successive tracer
atom jump directions comes about purely
because of the proximity of the defect. Un-
like the vacancy diffusion mechanism,
however, in this case the correlation comes
about onlyfor consecutive pairs of tracer
jumps of the type interstitial site Æregular
lattice site Æinterstitial site.
In the presence of correlation between
jump directions, the sum of the dot prod-
ucts ·r
i·r
i+jÒno longer averages out to
zero. A convenient way of expressing these
correlations quantitatively is to form the
ratio of the actual ·R
2
Òto the ·R
2
Òresult-
ing from a complete random walk, i.e.,
·R
2
Ò
random:
(3-73)
where the quantity fis called the tracer cor-
relation factor or simply the correlation
factor. Sometimes the terminology “corre-
lation coefficient” is used, but this is to be
discouraged. With Eqs. (3-71) and (3-72)
we have
(3-74)
f
n
i
n
j
ni
iij
i
n
i
=
==
=
lim
→∞
−−
+
+
〈⋅ 〉
〈〉
∑∑
∑
1
2
1
1
1
1
2
rr
r
f=
random
〈〉
〈〉
R
R
2
2
It should be noted that the limit nƕis
applied to Eq. (3-74) to ensure that all pos- sible correlations are included.
There have been numerous publications
concerned with the calculation of the tracer correlation factor. The earlier work has been reviewed in detail by Le Claire (1970), more recent work has been covered by All- natt and Lidiard (1993). Table 3-1 gives some values of ffor various mechanisms
and lattices. It should be noted from Table 3-1 that the correlation factor for the inter- stitial mechanism is unity. For this mecha- nism the interstitials, which are considered to be present at a vanishingly small concen- tration, move on an uncorrelated random walk, very much like the vacancy in the va- cancy-assisted diffusion mechanism. Accord- ingly, the second term in Eq. (3-71) drops out and f=1. If the interstitial concentration is
increased, the interstitials impede one an- other and correlations are introduced; as a result, the tracer correlation factor de- creases from unity. In fact, it continues to decrease until only one interstice is left va- cant. The situation now is identical to that for vacancy-assisted diffusion. The varia- tion of fwith vacancy (or vacant interstice)
concentration in the f.c.c. lattice is shown in Fig. 3-9. Of course, in the unphysical sit- uation where the interstitials do not “see” one another, i.e., multiple occupancy of a site is permitted, the interstitials continue to move on a complete random walk at all concentrations, and falways equals unity.
For cubic lattices all the jumps are of the
same length. Then we have that |r
i|=r:
(3-75)
Then we have, with Eqs. (3-68), (3-71), and (3-75)
(3-76)
D
nr f
t
rf
*= =
22
66
G
i
n
i
nr
=
=
1
22
∑〈〉r
194 3 Diffusion Kinetics in Solidswww.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

247. 3.3 Microscopic Diffusion 195
where Gis the jump frequency. The jump
frequency is further discussed in Sec. 3.3.4.
This partitioning of the diffusion coeffi-
cient into its uncorrelated (
Gr
2
/6) and cor-
related (f) parts is very basic to random
walk theory. Other partitionings are cer-
tainly possible, e.g., only jumps which are
not immediately cancelled contribute to the
jump frequency, i.e., an “effective jump”
frequency, but the partitioning here seems
to be the most natural.
The tracer correlation factor itself can be
expressed as
(3-77)
This equation has been very useful for di-
rect computer simulation calculations of f;
see the review by Murch (1984a).
We see that fnormally acts to decrease
the tracer diffusion coefficient from its ran-
dom walk (f=1) value. The inclusion of f
in the expression for D* is necessary for a
complete description of the atomic diffu-
sion process. From Table 3-1, however, it
can be seen that for many 3D lattices f
really only decreases D* by some 20–
30%. This is not much more than the preci-
sion routinely obtainable in measurements
of the tracer diffusion coefficient (Roth-
man, 1984) and, given the difficulty in cal-
culating the jump frequency
G(see Sec.
3.3.4), may not appear to be particularly
significant. There are, however, many rea-
sons why a study of correlation effects in
diffusion is sufficiently important that it
has consumed the energies of many re-
searchers over almost a 40-year period.
f
R
nr
n
= lim
→∞
〈〉
2
2
Table 3-1.Some correlation factors (at infinitely low defect concentrations) from Le Claire (1970), Manning
(1968), and Murch (1982d).
Lattice Mechanism f
Honeycomb Vacancy 1/3
Square planar Vacancy 1/(p–1)
Triangular Vacancy 0.56006
Diamond Vacancy 1/2
B.c.c. Vacancy 0.72714
Simple cubic Vacancy 0.65311
F.c.c. Vacancy 0.78146
F.c.c. Divacancy 0.4579 ± 0.0005
All lattices Interstitial 1
NaCl structure Collinear interstitialcy 2/3
CaF
2structure (F) Non-collinear interstitialcy 0.9855
CaF
2structure (Ca) Collinear interstitialcy 4/5
CaF
2structure (Ca) Non-collinear interstitialcy 1
Figure 3-9.Tracer correlation factor vs. vacancy
concentration for non-interacting vacancies in the
f.c.c. lattice; after Murch (1975).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

248. First, as can be seen from Table 3-1, the
tracer correlation factor is quite sensitive to
the mechanism of diffusion operating. Al-
though f, by itself, is not measurable, mea-
surements of the Haven ratio (see Sec.
3.3.2)and the isotope effect (see Sec. 3.3.3)
(which are closely related to f) in favorable
cases can throw considerable light on the
mechanism(s) of diffusion that are operat-
ing. Identification of the diffusion mecha-
nism is surely the most important ingredi-
ent in understanding the way atoms mi-
grate in solids and how it can be controlled.
Much of our discussion so far has been
concerned with pure solids with few de-
fects. There are, however, a large number
of solids where the apparent defect concen-
tration can be fairly high, e.g., highly non-
stoichiometric compounds, fast ion con-
ductors, and certain concentrated intersti-
tial solid solutions. In such cases the de-
fects interact and correlation effects tend to
be magnified and become highly tempera-
ture dependent. Then the apparent activa-
tion energy for tracer diffusion includes an
important contribution directly from the
correlation factor. Clearly, an understand-
ing of f is a very important part of the
understanding of the diffusion process in
such materials. We shall deal further with
this subject in Sec. 3.3.1.6. In ordered
binary alloys, the correlation factor can
become very small since atoms which
make a jump from the “right” lattice to the
“wrong” lattice tend to reverse, i.e., cancel
that jump. As a result, the tracer diffusion
coefficient can become abnormally small,
largely because of this strong memory ef-
fect. We shall deal further with this subject
in Sec. 3.3.1.5.
Another important case of correlation ef-
fects is impurity diffusion. The correlation
factor for the impurity is highly dependent
on the relative jump frequencies of the im-
purity and host atoms in the vicinity of the
impurity. We shall deal with this subject in
the following section.
3.3.1.4 Impurity Correlation Factor
We first consider that the impurity con-
centration is sufficiently dilute that there is
not a composition-dependent impurity dif-
fusion coefficient. Accordingly, each im-
purity atom is considered to diffuse in pure
host. The jump frequency of the impurity is
given by
w
2whereas the jump frequency of
the host atoms is given by
w
0. The signifi-
cance of the subscripts will be apparent
later. The vacancy mechanism is assumed.
The “tracer” now is the impurity. The
result for the impurity correlation factor f
2
in the f.c.c. lattice as a function of w
2/w
0
is shown in Fig. 3-10. When w
2<w
0(upper
curve), the impurity motion becomes
considerably decorrelated since, after an
impurity/vacancy exchange, the vacancy
does not trend to remain in the vicinity of
the impurity. When the impurity is next
196 3 Diffusion Kinetics in Solids
Figure 3-10.Impurity correlation factor in the f.c.c.
lattice as a function of
w
2/w
0: points by computer
simulation (Murch and Thorn, 1978), solid lines
from the formalism of Manning (1964); after Murch
and Thorn (1978).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

249. 3.3 Microscopic Diffusion 197
approached by the vacancy, it will tend
to be from a random direction, with the
result that the impurity correlation factor
f
2Æ1.0, thereby signifying a less corre-
lated random walk.
Conversely, when
w
2>w
0(lower curve),
the impurity and vacancy tend to continue
exchanging places in a particular configu-
ration. Many impurity jumps are thereby
effectively cancelled and f
2Æ0.0, i.e.,
the jumps are more correlated. Of course,
when
w
2=w
0the “impurity” is now a
tracer in the host, and in effect it can be
considered to have a different “color” from
the host. The impurity correlation factor
now is the tracer correlation factor, which
in this example is 0.78146 (see Table 3-1).
The diffusion coefficient can be consid-
ered to be the product of an uncorrelated
and a correlated part (Eq. (3-76)). In this
example, when
w
2>w
0, the reduced im-
purity correlation factor acts to reducethe
impurity diffusion coefficient from that ex-
pected on the basis of
w
2alone. Similarly,
when
w
2<w
0the raised impurity correla-
tion factor acts to increase the impurity dif-
fusion coefficient from that expected on
the basis of
w
2alone.
In general, the presence of the impurity
atom in the host also influences the host
jump frequencies in the vicinity of the im-
purity so that they differ from
w
0. For the
f.c.c. lattice the usual model adopted is the
so-called five-frequency model and is de-
picted in Fig. 3-11. The frequency
w
1is
the host frequency for host atom/vacancy
jumps that are bothnearest neighbors to the
impurity. This jump is often called the “ro-
tational jump” because in effect the host can
rotatearound the impurity. The frequency
w
3refers to a “dissociative” jump, i.e., a
host atom jump which takes the vacancy
away fromthe impurity. Finally
w
4(not
shown) is the “associative” jump, which is
a host jump that brings the vacancy to the
nearest neighbor position of the impurity,
i.e., the reverse of
w
3. All other host jumps
occur with the frequency
w
0.
We have mentioned “associative” and
“dissociative” jumps of the vacancy. These
are directly related to the impurity–va-
cancy binding energy E
Bat the first nearest
neighbor separation by
w
4/w
3= exp (–E
B/kT) (3-78)
Note that E
Bis negative for attraction
between the impurity and the vacancy.
Manning (1964) has shown that the im-
purity correlation factor f
2is given rigor-
ously for the five-frequency model by
f
2= (2w
1+ 7Fw
3)/(2w
2+ 2w
1+ 7Fw
3)
(3-79)
where Fis the fraction of dissociating va-
cancies that are permanently lost from a
site and are uncompensated for by return-
ing vacancies. Fis given to a very good ap-
proximation by
and
a=w
4/w
0.
77
10 180 5 927 1341
2 40 2 254 597 436
432
432
F=(3-80)
.
.
−
+++
++++
⎡
⎣
⎢
⎤
⎦
⎥
aaaa
aaaa
Figure 3-11.Five-frequency model for impurity dif-
fusion in the f.c.c. lattice (see text), figure taken from
Murch and Thorn (1978).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

250. Other more accurate expressions for F
have been reviewed by Allnatt and Lidiard
(1987, 1993). For many practical purposes
the extra rigor is probably unwarranted.
Extension to the case where there are va-
cancy–impurity interactions at second
nearest neighbors has been considered by
Manning (1964).
In the b.c.c. lattice the second nearest
neighbor is close to the nearest neighbor
(within 15% in fact). In a formal sense we
need to consider dissociativejumps for a
vacancy, jumping from the first to the sec-
ond, third (not the fourth), and fifth nearest
neighbors: these are denoted by
w
3, w¢
3,
and
w≤
3.
The corresponding reverse associative
jumps are
w
4, w¢
4, and w≤
4. We also need to
consider dissociative jumps from the sec-
ond nearest neighbor to the fourth (
w
5) and
the reverse of this (
w
6). There is no rota-
tional, i.e.,
w
1, jump.
The binding energy E
B1between the im-
purity and the vacancy at first nearest
neighbor is expressible as (see, for exam-
ple, Bocquet et al., 1996)
w¢
4/w¢
3= w≤
4/w≤
3= exp (–E
B1/kT) (3-81)
The binding energy E
B2between the im-
purity and the vacancy at second nearest
neighbors can be expressed as
w
6/w
5= exp (–E
B2/kT) (3-82)
We also have that
w
4w
4/w
5w
3= w¢
4/w¢
3 (3-83)
In order to proceed it has been usual to re-
duce the large number of jump frequencies
by making certain assumptions. There are
two basic models. In the first, usually
called Model I, it is assumed that
w
6=w
0=
w¢
4=w≤
4. This implies that w¢
3=w≤
3and
w
3w
5=w¢
3w
4. In the second, Model II,
on the other hand, the impurity–vacancy
interaction is in effect restricted to the first
nearest neighbor by means of the assump-
tions
w
3=w¢
3=w≤
3and w
5=w
6=w
0. This
also implies that
w
4=w¢
4=w≤
4.
Model I leads to the expression (Man-
ning, 1964; Le Claire, 1970)
(3-84)
with F given by
(3-85)
and
a=w
3/w¢
4.
Model II leads to the expression
(3-86)
where F is given by
(3-87)
and
a=w
4/w
0.
Correlation factors for impurity diffu-
sion via vacancies in the simple cubic lat-
tice (Manning, personal communication,
cited by Murch, 1982a) and the diamond
structure have also been calculated (see
Manning, 1964), as well as the h.c.p. struc-
ture (see Huntington and Ghate (1962) and
Ghate (1964)). Correlation factors for im-
purity diffusion by interstitialcy jumps
have also been reported for the f.c.c. lattice
and the AgCl structure (Manning, 1959).
In many cases of impurity diffusion the
expression for the impurity correlation fac-
tor is of the form
(3-88)
where ucontains onlythe host frequencies;
e.g., for vacancy diffusion in the f.c.c. lat-
tice u is given by
(3-89)
uF=w
w
ww
1
3
40
2
+ (/)
f
u
u
2
2=
w+
7
3 3343 9738 6606
8 68 18 35 9 433
32
32
F=
aaa
aa a+++
+++
...
...
f
F
F
2
3
23
7
27
=w
ww
+
7
2 5 1817 2 476
0 8106
2
F=
aa
a++
+
..
.
f
F
F
2
3
23
7
27
=
′
+′w
ww
198 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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251. 3.3 Microscopic Diffusion 199
This form for the impurity correlation fac-
tor is especially relevant to the isotope ef-
fect in diffusion (see Sec. 3.3.3) where for
self-diffusion the isotopes are tracers of the
host but, being isotopes, actually have
slightly different jump frequencies from
the host and can be classified as “impur-
ities”. We might also mention that for im-
purity diffusion experiments the isotopes
can additionally be of the impurity.
As a closing remark for this section, it
should be remembered that because of its
mathematical form, the impurity correla-
tion factor will be temperature dependent.
Over a fairly small temperature range the
temperature dependence can frequently be
approximated by an Arrhenius-like expres-
sion, i.e.,
f
2≈f
2
0exp (–Q¢/kT) (3-90)
where Q¢is some activation energy for the
correlation process (this has no particular
physical meaning), kis the Boltzmann con-
stant, and f
2
0is a “constant”. It should be
noted that Q¢will be unavoidably included
in the activation energy for the entire im-
purity diffusion process and is not neces-
sarily unimportant.
3.3.1.5 Correlation Factors for
Concentrated Alloy Systems
The impurity correlation factors dis-
cussed in the previous section are probably
appropriate for impurity concentrations up
to about 1 at.%. At higher concentrations
the impurities can no longer be considered
independent in the sense that the correla-
tion events themselves are independent.
This presents a special difficulty because
there is no easy way of extending, say, the
five-frequency model into the concentrated
regime without rapidly increasing the num-
ber of jump frequencies to an unworkable
number. As a result, models have been in-
troduced which limit the number of jump
frequencies but only as a result of some
loss of realism.
The first of these is the “random alloy”
model introduced by Manning (1968,
1970, 1971). The random alloy is of con-
siderable interest because, despite its sim-
plicity, it seems to describe fairly well the
diffusion behavior of a large number of al-
loys. The atomic components (two or
more) are assumed to be ideally mixed and
the vacancy concentration is assumed to be
very low. The atomic jump frequencies,
e.g.,
w
Aand w
B, in a binary alloy are expli-
citly speficied and neither changes with
composition or environment. In the very
dilute limit this model formally corre-
sponds, of course, to specifying a host
jump frequency
w
0and an impurity jump
frequency
w
2with all other host jump fre-
quencies equal to
w
0. However, it should
be appreciated for this model that physi-
cally, although not mathematically, the
jump frequencies
w
Aand w
Bare conceived
to be “average” jump frequencies. The
average jump frequency of the vacancy,
w
v,
was postulated to be given by
w
v= c
Aw
A+ c
Bw
B (3-91)
where c
Aand c
Bare the atomic fractions
of A and B. Manning (1968, 1970, 1971)
finds that
(3-92)
where
S= {[(c
B– x
0)w
B/w
A+ (c
A– x
0)]
2
+ 4x
0(1 – x
0)w
B/w
A}
1/2
(3-92a)
and
x
0= 1 – f (3-92b)
and fis the correlation factor for vacancy
diffusion in the pure metal A or B.
f
Sxc cx
Sx x
A
BBA A
BA=
+− + + −
+− +
()/()
{( )/ }
23
21
00
00ww
wwwww.iran-mavad.com
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252. Early Monte Carlo calculations indi-
cated that Eq. (3-92) was a very good ap-
proximation over a wide range of
w
A/w
B
but more recently it has been found that
this was largely illusory (Belova and
Murch, 2000). Eq. (3-92) is actually only a
reasonable approximation when the ex-
change frequency ratio is within about an
order of magnitude of unity. A much better
approximation to the problem is that given
by Moleko et al. (1989), but unfortunately
the equations are much more cumbersome
to use.
A different approach to diffusion in con-
centrated systems has been taken by Kiku-
chi and Sato (1969, 1970, 1972). They de-
veloped their Path Probability Method
(PPM) to cope specifically with the prob-
lems of diffusion in concentrated systems.
The method can be considered to be a time-
dependent statistical mechanical approach.
They started with the so-called binary alloy
analogue of the Ising antiferromagnet
model. In this model, sometimes called the
“bond” model, interactions E
AA, E
BB, and
E
ABare introduced between nearest neigh-
bor components of the type A-A, B-B, and
A-B, respectively. Equilibrium properties
of this model are well known; see for ex-
ample, Sato (1970). It is convenient to fo-
cus on the ordering energy Edefined by
E= E
AA+ E
BB– 2E
AB (3-93)
(note that in the literature there are other
definitions of the ordering energy which
differ from this one by either a negative
sign or a factor of 2). When E> 0, there is
an ordered region in the b.c.c. and s.c. lat-
tices (the f.c.c. lattice is more complicated
but has not been investigated by the PPM).
The ordered region is symmetrically placed
about c
A= 0.5. When E < 0, there is a two-
phase region at lower temperatures (this
side has not been investigated by the PPM).
The exchange frequencies
w
Aand w
Bare
not given explicitly as in the random alloy
model but are expressed in the following
way in terms of the interaction energies
(bonds) for a given atom in a given config-
uration:
w
i= n
iexp (–U
i/kT)
¥exp [(g
iE
ii+ g
jE
jj)/kT], (3-94)
i= A, Bi(j
where E
ij< 0, n
iis the lattice vibration fre-
quency, U
iis the reference saddle point en-
ergy in the absence of interactions, g
iis the
number of atoms of the same type which
are nearest neighbors of the given atom and
g
jis the number of atoms of the other type
which are nearest neighbors of the given
atom. Other forms of
w
iare also possible
but have not been developed in this con-
text.
In the PPM a path probability function
is formulated and maximized, a process
which is said to be analogous to mini-
mizing the free energy in equilibrium sta-
tistical mechanics. The first calculations
were made using what was called a time-
averaged conversion. The results were in
poor agreement with later Monte Carlo
computer simulation results. Subsequently,
substantial improvements were made
(Sato, 1984) and the PPM results are now
in fairly good agreement with computer
simulation results. A typical comparison of
the earlier results with Monte Carlo results
from Bakker et al. (1976) is shown in Fig.
3-12. Note the bend at the order/disorder
transition. The lower values of the correla-
tion factors on the ordered side can be
ascribed generally to the higher probability
of jump reversals as an atom which jumps
from the “right” sublattice to the “wrong”
sublattice tends to reverse the jump while
the vacancy is still present. This cancella-
tion of jumps obviously leads to small val-
ues of f.
200 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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253. 3.3 Microscopic Diffusion 201
Although there is no particular reason
for assuming that the correlation factor
should follow an Arrhenius behavior, e.g.,
Eq. (3-90), these results nonetheless show
it quite well. The contribution to the total
activation energy is difficult to determine
directly because U
iis a “spectator quan-
tity” in these calculations and the vacancy
formation energy has not been calculated
in the same work, although the latter can be
determined separately in this model (Lim
et al., 1990). In a recent application of this
model to self-diffusion in
b-CuZn (Belova
and Murch, 1998), the contribution to the
totalactivation energy for, say, Cu tracer
diffusion was variously estimated at 22–
40%. It is clear that for a complete analysis
of diffusion in such materials the assess-
ment of the contribution of the tracer corre-
lation factor to the activation energy is es-
sential.
We have mentioned that cancellation of
jumps leads to small values of f. There are,
however, special sequences of jumps in the
ordered structure which lead to effective
diffusion. The most important of these se-
quences is the so-called six-jump cycle
mechanism, which we shall now briefly
discuss. This sequence is interspersed, of
course, with jump reversals.
It became clear at an early stage that dif-
fusion in fully ordered structures poses cer-
tain difficulties. A purely random walk of
the vacancy will lead to large amounts of
disorder and inevitably to a great increase
in the lattice energy of the solid. Hunting-
ton (see Elcock and McCombie, 1958)
seems to have been the first to suggest that net
migration of atoms in, say, the B2 structure
could occur by way of a specific sequence
of six vacancy jumps with only a relatively
slight increase in energy. This sequence is
illustrated in Fig. 3-13a for the ordered
square lattice (Bakker, 1984), and sche-
matic energy changes for the sequence are
shown in Fig. 3-13b. The starting and fin-
ishing configurations have the same energy
but there has been a net migration of atoms.
The correlation factor for the six-jump
cycle sequence alone has been calculated
by Arita et al. (1989). The six-jump cycle
sequence is contained in the earlier path
probability method calculations and the
Monte Carlo computer simulations (Bak-
ker, 1984). The correlation factors calcu-
lated in those calculations are statistical av-
erages over all possible sequences in addi-
tion to the jump reversals which tend to
predominate.
Further detailed discussion of the corre-
lation factors in concentrated alloy systems
can be found in the detailed reviews by
Bakker (1984), Mehrer (1998), Murch and
Belova (1998).
3.3.1.6 Correlation Factors for
Highly Defective Systems
Many solids, such as nonstoichiometric
compounds, intercalation compounds, and
fast ion conductors, appear to have a very
high concentration of defects. Manning
Figure 3-12.Arrhenius plot of the tracer correlation
factor for B in the alloy A
0.6B
0.4for various values of
U=(E
AA–E
BB)/E. Points from Bakker et al. (1976),
lines from Kikuchi and Sato (1969, 1970, 1972); af-
ter Murch (1984a). The abscissa is in units of E/k.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

254. (1968) was probably the first to note that the
correlation factor could increase if more
vacancies are present. In effect, the extra
vacancies decorrelatethe reverse jump of a
tracer atom; see the discussion in Sec.
3.3.1.3. As more vacancies are added we
reach the limit of a single atom remaining
(for convenience, the tracer) and the corre-
lation factor is unity as befitting a complete
random walk of an interstitial. Between the
extremes, provided the vacancies are ran-
domly distributed, roughly linear behavior
of the correlation factor is found (Fig. 3-9).
When the vacancies are notrandomly
distributed, which is usually the case be-
cause of atom–atom repulsion, the behav-
ior of the correlation factor becomes more
complicated. The calculations that have
been performed are for the so-called lattice
gas model. In this, the atoms occupy dis-
crete sites of a rigid lattice, typically a
nearest neighbor interaction being speci-
fied between the atoms. For nearest neigh-
bor attractiveinteractions, at low tempera-
tures a two-phase region develops symmet-
rically about the concentration 0.5. Con-
versely, with nearest neighbor repulsion an
ordered region centered about a concentra-
tion 0.5 develops for the simple lattices
such as honeycomb, square planar, or sim-
ple cubic. The face centered cubic and tri-
angular lattices are rather more compli-
cated, but have not been investigated. We
shall focus on repulsion here because it is
the most likely. In a manner similar to that
for the concentrated alloy, the exchange
frequency of a given atom with a vacancy
is written for a given configuration as
w=nexp (–U/kT)exp(g
nnf
nn/kT) (3-95)
where g
nnis the number of atoms which are
nearest neighbors to the given atom, Uis
the activation energy for diffusion of an
isolated atom,
nis the vibration frequency,
and
f
nnis the atom–atom interaction en-
ergy. It can be seen that nearest-neighbor
repulsion between atoms works to diminish
the activation energy. Other forms of the
exchange frequency are also possible, but
most calculations have used this one.
Obviously lattice gas models cannot
generally be very realistic in their diffusion
behavior. The interest in them comes about
primarily because they exhibit behavior
which is rich in physics and likely to occur
202 3 Diffusion Kinetics in Solids
Figure 3-13.(a) Six-jump cycle in the ordered square
lattice: the upper figure shows the path of the va-
cancy, the lower figure shows the displacements re-
sulting from the complete cycle, after Bakker (1984).
(b) Schematic representation of energy changes dur-
ing the six-jump vacancy cycle, after Bakker (1984).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

255. 3.3 Microscopic Diffusion 203
to a greater or lesser extent in real materi-
als. Surprisingly, some materials such as
interstitial solid solutions are described
fairly well by the lattice gas model even
with only nearest neighbor interactions.
This must be a result of short-range inter-
actions being by far the most important.
Sato and Kikuchi (1971) have employed
the PPM to good effect in the lattice gas
model. Extensive calculations have been
made, especially for the honeycomb lattice.
Equally extensive Monte Carlo calcula-
tions have also been made both for this lat-
tice and many others; see the review by
Murch (1984a). A typical result is shown
in Fig. 3-14. The pronounced minimum in
the correlation factor is due to the effects of
the ordered structure. This ordered struc-
ture consists of atoms and vacant sites ar-
ranged alternately. The wide composition
variability of the ordered structure at this
temperature (indicated by the arrows) is ac-
commodated by vacant sites or “intersti-
tials” as appropriate. When an atom jumps
from the right lattice to the wrong lattice it
tends to reverse that jump, thereby giving
low values of the correlation factor. As in
the ordered alloy, there are sequences of
jumps which lead to long-range diffusion.
By and large these sequences are based
around interstitialcy progressions, but
rather than two atoms moving in unison as
in the actual interstitialcy mechanism, here
they are separated in time.
The temperature dependence of fin the
lattice gas, like the alloy, is fairly strong.
Again, it is usual to write fin an Arrhenius
fashion, e.g., Eq. (3-90), but there is no
known physical reason for assuming that f
must really take this form. Fig. 3-15 shows
the behavior in the square planar lattice
gas. The Arrhenius plot is in fact curved
above and below the order/disorder tem-
perature, though in an experimental study
this would be overlooked because of a
much smaller temperature range. This acti-
vation energy will be included in the ex-
perimental tracer diffusion activation en-
ergy. This is likely to be a significant con-
tribution which (like in the ordered alloy)
cannot be ignored.
The subject of tracer correlation effects
in defective materials with high defect con-
Figure 3-14.PPM results for the dependence of the
tracer correlation factor on ion site fraction c
iin the
honeycomb lattice with nearest neighbor repulsion.
T*=kT/
f
nn; after Murch (1982d).
Figure 3-15.Arrhenius plot of the tracer correlation
factor gained from Monte Carlo simulation of a square planar lattice gas with 50% particles and 50% vacant sites (Zhang and Murch, 1990).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

256. tent is dealt with at length in a number of
places; see, for example, Murch (1984a)
and Sato (1989).
3.3.1.7 The Physical or Conductivity
Correlation Factor
In Sec. 3.3.1.3 we have seen that tracer
correlation effects are conventionally em-
bodied in the so-called tracer correlation
factor. In this way the correlation factor ap-
pears as a correction factor in the random
walk expression for the tracer diffusion co-
efficient Eq. (3-76).
In 1971 Sato and Kikuchi showed that
the ionic conductivityshould also include a
correlation factor, now called the physical
or conductivity correlation factor and
given the symbol f
I, sometimes f
c. In the
usual hopping model expression for the
d.c. conductivity
s(0), we have
s(0) = C(Ze)
2
G
qr
q
2f
I/6kT (3-96)
where Cis the concentration of charge car-
riers per unit volume, Zis the number of
charges, eis the electronic charge,
G
qis the
jump frequency, r
qis the jump distance of
the charge, kis the Boltzmann constant,
and T is the absolute temperature.
It should be noted that f
Idoes not be-
come nontrivial, i.e., 71, until a relatively
high defect concentration is present and
ion–ion interactions and/or trapping sites
are present. A model containing these was
explored by Sato and Kikuchi (1971) in
their pioneering work. Their results for f
Iin
the lattice gas model of nearest neighbor
interacting atoms diffusing on a honey-
comb lattice with inequivalent sites ar-
ranged alternately are shown in Fig. 3-16.
This work was the starting point for the
development of the new area of correla-
tion effects in ionic conductivity. This area
has been reviewed by Murch and Dyre
(1989).
Historically, the atomistic meaning of f
I
has taken quite some time to determine,
principally because the calculations (path
probability and Monte Carlo methods)
were based on the calculation of a flow of
charge.
It is now known that the physical corre-
lation factor can be expressed as
(3-97)
where DRis the total displacement of the
systemafter a total of njumps in time t.
This differs from Eq. (3-77) onlyin the im-
portant points that DRrefers to the dis-
placement of the entire system and nota
single particle. DRis simply the sum of the
individual particle displacements Roccur-
ring in time t:
(3-98)
We can now see that whereas fencom-
passes correlation effects of a single (tracer)
DRR=
all
particle
∑ i
fn r
n
I
= lim /
→∞
〈〉DR
22
204 3 Diffusion Kinetics in Solids
Figure 3-16.PPM results for the dependence of the
physical correlation factor on ion site fraction c
iin
the honeycomb lattice with nearest neighbor repul-
sion and site inequivalence. T*=kT/
f
nnand the site
difference a priori is 5
f
nn; after Murch (1982d).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

257. 3.3 Microscopic Diffusion 205
particle, f
Iencompasses correlation effects
of the entire system in a collective sense.
The correlation factor f
Iof the system is
manifested in the d.c. ionic conductivity,
Eq. (3-96). It is also manifested in chemi-
cal diffusion (see below), but notin tracer
diffusion.
What is sufficient to make f
I71? First,
a relatively high concentration of defects
is required (see Fig. 3-16). Next, inequality
of lattice sites (and therefore a variable
jump frequency), or mutual interactions
among the ions, or differences in acces-
sibilities of ions: all are sufficient, with a
high concentration of defects, to give a
nontrivial value forf
I.
The importance of f
Iin materials with
high defect concentrations cannot be over-
stated. For example, in a calculation of f
I
in a model for CeO
2doped with Y
2O
3
(to obtain a high concentration of anion va-
cancies), it was found that at 455 K f
I
changed from unity at low Y
3+
content to
only 0.05 at high Y
3+
content, correspond-
ing to 14% of the anion sites being vacant
(Murray et al., 1986). This very low value
of f
Icame about primarily because of the
trapping effects of the Y
3+
ions. The phys-
ical correlation factor is also temperature
dependent, but there is no known physical
reason why f
Ishould take an Arrhenius
form. Fig. 3-17 shows a typical result for a
lattice gas above and below the order/dis-
order transition. Both regions are slightly
curved in fact, although this could be over-
looked in an experiment over a small tem-
perature range. The activation energy asso-
ciated with f
Iwill be included in the overall
activation energy for d.c. ionic conduction.
This contribution cannot be ignored. In the
model for CeO
2doped with Y
2O
3, the con-
tribution is of the order of 30% very high
Y
3+
contents.
Most of the contributions to the applica-
tion of f
Ihave centered around ionic con-
duction. However, f
Ialso occurs in the ex-
pression for the chemical diffusion coeffi-
cient in the one-component system, i.e.,
D˜=
Gr
2
f
I(1 + ∂ lng/∂lnc)/6 (3-99)
or, with Eq. (3-76)
where
gis the activity coefficient of the
particles with the term in parentheses be-
coming the “thermodynamic factor”. This
equation applies to interstitial solid solu-
tions and intercalation compounds such as
TiS
2intercalated with Li where the atomic
mobility on the defective lattice is rate
determining. It does notapply to ionic
conductors, however, since – for change
balance reasons – compositionalchanges
are controlled by atomic mobility or hole-
electron hopping on another lattice. For
example, in the oxygen ion conductor
calcia stabilized zirconia, compositional
changes of the oxygen ion vacancy concen-
tration are probably controlled by the
dopant calcium ion mobility on the cation
lattice.
=
I
D
f
f
c
*ln
ln
1+
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟ g
Figure 3-17.Arrhenius plot of the physical correla-
tion factor gained from Monte Carlo simulation of a
square planar lattice gas with 50% particles and 50%
vacant sites (Zhang and Murch, 1990).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

258. 3.3.1.8 Correlation Functions
(Collective Correlation Factors)
There are a number of other correlation
phenomena in diffusion. In order to discuss
these in a unified fashion, say for a binary
system, it is convenient first to introduce
correlation functions or collective correla-
tion factors. They represent the correlated
parts of the phenomenological coefficients
(see Sec. 3.4). For the diagonal coefficients
we write (Allnatt and Allnatt, 1984)
L
ii= L
0
ii
f
iii= A, B (3-100)
where L
0
ii
is the uncorrelated part of L
ii,
i.e.,
L
ii
(0)= C
iG
ir
2
/6kT i= A, B (3-101)
where C
iis the number of species iper unit
volume,
G
iis the jump frequency of species
i, and ris the jump distance, kis the Boltz-
mann constant, and Tis the absolute tem-
perature. For the off-diagonal terms (we
note the Onsager condition L
AB=L
BA), we
write
L
AB= L
(0)
AA
f
AB
(A)= L
BB
(0)f
AB
(B) (3-102)
This partitioning of the L
ijinto these parts
might appear arbitrary. The choice comes
about largely by analogy with the tracer
diffusion coefficient which is partitioned as
a product of an uncorrelated part, contain-
ing the jump frequency etc., and a corre-
lated part, containing the tracer correlation
factor (see Eq. (3-76) and associated dis-
cussion).
Some physical understanding of the f
ijis
appropriate here. There are two equivalent
expressions for the f
ij. First, Allnatt (1982)
and Allnatt and Allnatt (1984) showed that
the f
ijhave Einsteinian forms reminiscent
of the expression for the tracer correlation
factor (see Eq. (3-76)). Thus they write for
a binary system A, B
f
AA= ·DR
2
A
Ò/n
Ar
2
(3-103)
where DR
Ais the displacement of the
system of particles of type A, i.e., DR
Ais
the vector sum of the displacements of all
the atoms of type A and n
Ais the total
number of jumps of the species A. Thus the
group of particles of type A in the system is
treated itselflike a (hypothetical) particle,
and f
AAis its correlation factor.
Similarly,
f
AB
(i)= ·DR
A· DR
BÒ/n
Ar
2
i= A, B (3-104)
Thus f
AB
(i)expresses correlations between
the vector sum of the displacements of all
the atoms of type A with the corresponding
quantity for atoms of type B.
An alternative way of looking at the f
ij
is in terms of the drifts in a driving force.
Let us consider a “thought experiment”
for a binary system AB where an external
driving force F
dis directly felt onlyby
the A atoms. It is straightforward to show
(Murch, 1982a) that f
AAis given by
(3-105)
where ·X
AÒis the average drift distance of
the A atoms in the driving force and n
Ais
the number of jumps of a given A atom
in time t . However, although only the A
atoms feel the force directly, the B atoms
feel it indirectlybecause they are compet-
ing for the same vacancies as the A atoms.
Accordingly, there is also a drift of the
B atoms, smaller than the drift of the A
atoms. The correlation function f
AB
(i)is re-
lated to this drift by
(3-106)
where ·X
BÒis the average drift distance of
the B atoms.
f
kT X
Ftr
i
i
AB
B
d
=
()2
2
〈〉
G
f
kT X
Fn r
AA
A
dA=
2
2
〈〉
206 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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259. 3.3 Microscopic Diffusion 207
Calculation of the Correlation Functions
Calculations have been made for various
impurity models, the random alloy model
and interacting or (bond) models for alloys.
For impurity systems, the area has been re-
viewed exhaustively by Allnatt and Lidiard
(1987, 1993). Here we shall restrict our-
selves to a discussion of the results for the
five-frequency impurity model in the f.c.c.
lattice; see Fig. 3-11 and associated text for
details of the model.
Inspection of Eq. (3-103) indicates that
when the A atoms (say) are the infinitely
dilute impurities in B, i.e., c
AÆ0, the
system of atoms of type A reduces to a
single A atom. Accordingly the diagonal
correlation function f
AAreduces to the im-
purity correlation factor, f
A. We have al-
ready given Manning’s (1964) expression
for f
Afor the five-frequency model in the
f.c.c. lattice (see Eq. (3-78)). Manning
(1968) showed that the cross correlation
function f
AB
(A)can be calculated by a careful
analysis of the various impurity jump tra-
jectories in a field. He found that f
AB
(A)is
given by
where Fis the fraction of dissociating va-
cancies that are permanently lost from a
site and are uncompensated for by return-
ing vacancies and is given by Eq. (3-80).
Let us move on to the binary random
alloy model described in detail in Sec.
3.3.1.5; explicit expressions have been de-
rived by Manning (1968). He found that
f
AAis given by
(3-108)
ff cfM cf
k
kkkAA A A A A
=A,B
=12
0+
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥ ∑ww
ff
F
F
AB
A
A
= (3-107)
()
[()( )]
2
371 2
27
34 04
1
1
13
×
−− − −
+
−
www ww
ww
and
(3-109)
where M
0=2f(1 –f) and fis the tracer cor-
relation factor in the pure lattice of either
component.
The correlation functions have also been
calculated in interacting bond models (see
Sec. 3.3.1.5), especially with the PPM and
by computer simulation. Some results of
the latter are shown in Figs. 3-18 and 3-19
for the simple cubic lattice with ordering
between A and B.
A brief discussion of the behavior of
these quantities is appropriate here. First,
with respect to f
AA, as c
AÆ0, the A atoms
behave like impurities in the B matrix, and
f
AAdoes in fact reduce to the impurity cor-
relation factor f
A. At c
A≈0.5, the minima
are further manifestations of the prepon-
derance of jump reversals in the diffusion
process in the ordered alloy. As c
AÆ1, all
curves converge on unity and the correla-
fcc ff
cM cf
k
kkk
AB
A
AB A BAB
AA
=A,B
=
()
2
0
1
ww
ww
×
⎛
⎝
⎜
⎞
⎠
⎟ ∑
−
Figure 3-18.Monte Carlo results for the diagonal
correlation factor f
AAin the simple cubic alloy as a
function of c
Aat various values of the ordering en-
ergy; after Zhang et al. (1989a).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

260. tion effects contained in f
AAdisappear en-
tirely. Next, let us discuss f
AB
(A). As c
AÆ0,
again A behaves like an impurity in B. The
physical meaning of f
AB
(A)has been dis-
cussed earlier, see Eq. (3-106). At this limit
the B atoms have maximum interference
on A. Similar to f
AA, minima develop at
c
A≈0.5 as a result of jump cancellations.
As c
AÆ1, all curves converge on zero as
the B atoms, now in the minority, no longer
have much effect on the diffusion of A.
With the correlation functions in hand,
we can form the various correlation terms
that commonly occur in ionic conductivity
and chemical diffusion. We shall restrict
ourselves to binary systems. Unary sys-
tems are dealt with in Sec. 3.3.1.7, where it
is seen that the physical correlation factor
is really a correlation function for a one-
component system.
Correlations in Ionic Conductivity
(Binary Systems)
When ionic conductivity occurs in a ma-
terial where two or more ionic species
sharing the same sublattice (and therefore
competing for the same defects) carry the
current there are correlations or interfer-
ence between the two ionic conductivities.
When there are two ionic species and the
vacancy mechanism is operating, the ex-
pressions for the d.c. ionic conductivities
are (Murch and Dyre, 1989)
(3-110)
and
(3-111)
where eis the electronic charge, Z
A(B)is the
number of charges on A(B), C
A (B)is the
concentration of A(B),
G
A(B)is the jump
frequency of A(B) and ris the jump dis-
tance.
Sometimes the bracketed terms are called
the binary conductivity correlation factors;
they are, in fact, formally binary analogues
of f
Iin Eq. (3-96). Eqs. (3-110) and (3-111)
become
s
A= e
2
Z
2
A
C
AG
ir
2
f
IA/6kT (3-112)
s
B= e
2
Z
2
B
C
BG
ir
2
f
IB/6kT (3-113)
For those readers familiar with Man-
ning’s (1968) treatment of impurity ionic
conductivity, f
IA(where A is the impurity
in B) is expressed as
(3-114)
where f
Ais the impurity correlation factor
and ·n
pÒis a complex kinetic parameter
introduced by Manning. Eq. (3-114) as en-
visaged by Manning (1968) applies only to
situations where the vacancy concentration
is very low. Often the term in parentheses
in Eq. (3-114) is loosely called a vacancy-
wind factor, although this terminology is
frequently applied to the whole of f
IA. In
the special case where the impurity is a
ff
Z
Z
n
pIA A
B
A=1 +〈〉
⎛
⎝
⎜
⎞
⎠
⎟
s
B
BBB
BB
A
B
AB
B=
eZC r
kT
f
Z
Z
f
22 2
6
G
+
⎛
⎝
⎜
⎞
⎠
⎟
()
s
A
AAA
AA
B
A
AB
A=
eZC r
kT
f
Z
Z
f
22 2
6
G
+
⎛
⎝
⎜
⎞
⎠
⎟
()
208 3 Diffusion Kinetics in Solids
Figure 3-19.Monte Carlo results for the off-diago-
nal correlation factor f
A
AB
in the simple cubic alloy as
a function of c
Aat various values of the ordering en-
ergy; after Zhang et al. (1989a).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

261. 3.3 Microscopic Diffusion 209
tracer of the host and where, of course,
Z
A=Z
B, the term in parentheses reduces to
f
A
–1with the result that f
IA= 1 and there are
no correlations in the ionic conductivity.
However, it is emphasized that this result
applies onlywhere the defect concentration
is very low. For high concentrations of de-
fects which in general cannot be randomly
distributed there are still residual correla-
tions arising from non-ideal effects. These
are encompassed in f
I(see Eq. (3-96)).
For the five-frequency model, see Sec.
3.3.1.4, for certain combinations of the
jump frequencies it is possible to make f
IA
(Eq. (3-114)) negative. Specifically this
can occur when the vacancy and impurity
are tightly bound together, i.e., when
w
4>w
3. Vacancies that are bound to im-
purities can be transported around the im-
purity by the host atom flux. The probabil-
ity of the impurity moving upfield can then
be larger than the probability of moving
downfield. The impurity may then actually
move upfield, opposite to its “expected”
direction. For further details, see Manning
(1975). It is also possible in the five-fre-
quency model to make f
IAfor the impurity
exceed unity, even when Z
B=Z
A.
For concentrated interacting systems, the
calculation of the f
IA(and f
IB) is of some
interest, especially in the case of mixed fast
ion conductors such as Na,K
b-alumina.
An example of the results of calculation of
f
IAand f
IB, by way of the PPM, is shown in
Fig. 3-20. This lattice gas model contains
two species A and B (Z
A=Z
B) with 20%
vacancies on a honeycomb lattice. The
model approximates the fast ion conductor
Na,K
b-alumina, although it is probably
also a reasonable description of some sili-
cate glasses. Note the minima in f
IAand
f
IB. These are caused by what is called a
“percolation difficulty” in the flow created
by ordered arrangements. Physically, many
jumps are reversed. This effect seems
stronger as the dimensionality is lowered
(in fact all correlation effects do), and may
be a major contributor to the so-called
mixed-alkali effect. This effect is charac-
terized by a substantial decrease in the d.c.
ionic conductivity at intermediate mixed
compositions without any obvious physical
cause. This subject is dealt with further by
Murch and Dyre (1989) and Sato (1989).
Correlations in Chemical Diffusion
(Binary Systems)
The usual equation of practical interest
for chemical diffusion in a binary system is
the Darken equation. This equation relates
the intrinsic diffusion coefficient of a par-
ticular component to its tracer diffusion
coefficient. The original Darken equation
(which neglects correlations) is written as
(3-115)
DD
c
A
I
A*
A
A=1 +
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
ln
ln
g
Figure 3-20.PPM results for the quantities f
IAand
f
IBas a function of concentration r
Ain the two-com-
ponent conductor with 20% vacancies at half the or-
der–disorder temperature. The dashed line shows the
case where the development of long-range order is
artificially suppressed; after Murch and Dyre (1989).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

262. where g
Ais the activity coefficient of A.
However, it is straightforward to show that
the rigorous equation is Eq. (3-56) which
can be rewritten in terms of the correlation
functions (Le Claire, 1975; Murch, 1982a;
Stark, 1976):
(3-116)
The f
Ain the denominator is the correlation
factor for the actual composition of the al-
loy and not some impurity limit. In the lit-
erature the term derived from correlations
[ ] is sometimes called a vacancy-wind fac-
tor and is given the symbol r
A. As it turns
out, however, this term cannot vary much
from unity since the original Darken equa-
tion is reasonably well obeyed. The behav-
ior of r
Ais not especially transparent. The
vacancy flux in chemical diffusion is al-
ways in the same direction as that of the
slower moving species (and opposite to
that of the faster moving species). The va-
cancy-wind effect always tends to provide
a given atom with an enhanced probability
of flowing in a direction opposite to that of
the vacancy flow. When A is less mobile
than B, the vacancy-wind factor r
Aeffec-
tively makes D
I
A
even smaller. Conversely,
when A is more mobile than B, the va-
cancy-wind factor r
Aeffectively makes D
I
A
even larger. But how much? This can be
partially answered in the following.
Although we have given Manning’s
(1968) expressions for the individual f
ij
and the correlation factor f
Afor the random
alloy (Eqs. (3-108) and (3-109)), it may not
be immediately obvious that r
Ais in fact
given in this model by (see also Sec.
3.2.3.4)
(3-117)
r
fD c D D
fc D cD
A
B* A A* B*
AA* BB*=
+−
+
()
()
DD
c
ff
c
c
f
A
I
A*
A
A
AA AB
(A) A
B
A=1 +
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
×−
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
ln
ln
g
where fis the tracer correlation factor for
diffusion in the purecrystal, i.e., pure A or
pure B. It turns out that the maximum in r
A
in this modeloccurs when c
AÆ1. In the
extreme case when the jump frequency of
A is much larger than that of B, then
r
AÆf
0
–1. Therefore, when f
0= 0.78146 (va-
cancy diffusion on the f.c.c. lattice, see Ta-
ble 3-1), r
Acan only enhance chemical dif-
fusion by a factor of 1.28.
Manning’s equation (Eq. (3-117)) seems
to have more general significance than
application to the random alloy. Murch
(1982a) showed that this equation performs
very well indeed for an interacting alloy
model that can exhibit short and long-range
order. This suggests that this equation can
be used almost with impunity. This subject
arises again in the “Manning relations”
which relate the phenomenological coeffi-
cients to the tracer diffusion coefficients
(see Sec. 3.2.3.4).
The subject of the experimental check of
r
Ahas been discussed by Bocquet et al.
(1996). They pointed out that in most cases
the Manning formulation for r
Aand r
B
(Eq. (3-117)) improves the agreement be-
tween experimental and calculated values
of the Kirkendall shift and the ratio of
the intrinsic diffusion coefficients D
I
A
/D
I
B
.
However, the individual experimental val-
ues of D
I
A
and D
I
B
often tend to be quite a
bit higher than the calculated values. The
experimental Kirkendall shift also has a
tendency to be higher than the calculated
value. Carlson (1978) points out that the
problem could be due to the random alloy
assumption in the Manning formulation.
However, the success of the Manning for-
mulation for the interacting alloy model
mentioned above seems to vindicate the
random alloy assumption and so the reason
for the discrepancy probably should be
sought elsewhere.
210 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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263. 3.3 Microscopic Diffusion 211
3.3.2 The Nernst–Einstein Equation
and the Haven Ratio
The Nernst–Einstein equation relates
the d.c. ionic conductivity to a diffusion
coefficient. Probably no other equation in
diffusion has generated more misunder-
standing than this one. Let us consider the
standard derivation (see, for example,
Batchelor (1976)), and discuss its implica-
tions in detail.
We consider a pseudo-one-component
system in a situation where the flux result-
ing from an applied force on the particles
(which are completely noninteracting) ex-
actly counterbalances the flux due to diffu-
sion. That is, from Eq. (3-2)
(3-118)
It is important to note that the diffusion co-
efficient here refers to a chemical composi-
tion gradient and is most definitely con-
ceived as a chemical diffusion coefficient,
nota tracer or self-diffusion coefficient.
Lack of appreciation of this fact leads to
misunderstandings and inconsistencies.
The external force F
dis a result of a po-
tential so that
(3-119)
At equilibrium the distribution of com-
pletely noninteractingparticles follows a
Boltzmann distribution such that
C(x) = C
0exp [–f(x)/kT] (3-120)
Eq. (3-120) must be the solution of Eq. (3-
120) at steady state. We then have that
(3-121)
With Eq. (3-118) we find that
(3-122)
〈〉v
˜
D
F
kT
=
d
d
d
=
d
d
=
dC
x
C
kT C
CF
kT
− f
F
x
d=
d
d
−
f
〈〉vCD
C
x
=
d
d
˜
When the external force is the result of an electric field Ewe have
F
d= ZeE (3-123)
where Zis the number of charges (ionic
valence) and eis the electronic charge.
The mobility uis defined as the velocity
per unit field and so we have
(3-124)
In the solid-state diffusion literature Eq.
(3-124) is generally called the Nernst–Ein- stein relation. Because the d.c. ionic con- ductivity is related to the mobility by
s=
CZeu, Eq. (3-124) can be rewritten as
(3-125)
More generally, interactions are present between the particles and it can be shown that the generalform of the Nernst–Ein-
stein equation is in fact (Murch, 1982b)
(3-126)
where
mis the chemical potential of the
particles and cis the site fraction.
Now let us discuss this equation in detail
by exploring some particular cases. When the distribution of particles is completely ideal, meaning that the particles do not feel one another, not even site blocking, the thermodynamic factor drops out of Eq. (3- 126) and
s/D˜reduces to Eq. (3-125). In this
very special case, and only in this case, the tracer diffusion coefficient D* equals D˜, so
that
(3-127)
When the particles are ideally distributed but subject to the condition that no more than one particle can occupy one site, then
s
D
CZ e
kT*
=
22
s
m
˜
ln
D
CZ e
kT
c
=
22
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
s
˜
D
CZ e
kT
=
22
u
D
Ze
kT˜
=www.iran-mavad.com
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264. the thermodynamic factor in Eq. (3-126)
equals (1–c)
–1
. However, for this situation
the tracer diffusion coefficient is related to
D˜by (see, for example, Murch (1982c))
D*= D˜(1 –c)f (3-128)
where fis the tracer correlation factor, so
that
(3-129)
Ionic solids having virtually a perfect lat-
tice of particles (charge carriers) fall into
this category and Eq. (3-129) is appropriate
to such solids. Similarly, when the particles
are extremely dilute, Eq. (3-129) is again
appropriate.
In the solid-state diffusion literature we
very often see Eq. (3-125) used directly
to calculate another diffusion coefficient,
sometimes called the “charge” diffusion co-
efficient and given the symbol D
s. Thus we
encounter the following equation, which is
also called the Nernst–Einstein equation
(3-130)
D
sis dimensionally correct but it does not
correspond to any diffusion coefficient that
can actually be measured by way of Fick’s
laws for a solid system. Recall that the
identical Eq. (3-125) requires that the parti-
cles are completely noninteracting for it to
be meaningful. We frequently then see the
following equation relating D* and D
s
D*=fD
s (3-131)
with D
soften being called a self-diffusion
coefficient or the diffusion coefficient of
the (random walking) charge carriers. Eq.
(3-131) is generally given as if it is self-ev-
ident or similar to Eq. (3-73). Eqs. (3-130)
and (3-131) are then combined to give
(3-132)
s
D
CZ e
kT f*
=
22
s
sD
CZ e
kT
=
22
s
D
CZ e
kT f*
=
22
i.e., formally the same as Eq. (3-129). This route to Eq. (3-132) is clearly a case of two wrongs ending up making a right. How- ever, the real danger lies in the fact that Eq. (3-132) obtained in this way blinds us to its limitations, limitations which are clearly stated in the derivation of Eq. (3-129).
In practice, Eq. (3-132) has often been
used to describe situations where the par- ticles are interacting and many sites are vacant, such as in fast ion conductors. In such cases, we cannot necessarily expect a meaningful interpretation. What is the cor- rect way to proceed? We already have the Nernst–Einstein equation to cover the situ- ation of interacting particles (Eq. (3-126)). However, this normally cannot be applied to real materials because local charge neu- trality prevents composition gradients be- ing set up in ionic conductors by the con-
ducting ions themselves. What can be done
is to return once again to Eq. (3-130) and use it purely as a definitionof D
s, recognizing
at the same time that D
shas no Fickian
meaning. Eq. (3-129) is then being used purely as a means of changing
sto a quan-
tity which has the dimensions of a diffusion coefficient. It is clear that it would be inap- propriate in these circumstancesto call this
equation the Nernst–Einstein equation.
We can now define the Haven ratio,
which is simply the ratio of D* to D
s:
(3-133)
In view of what has been said above about D
s, it is appropriate to ask whether H
Rhas
any physical meaning. This can be partially answered by examining hopping models for conduction. It can be shown quite gen- erally for hopping models using the va- cancy mechanism that
(3-134)
H
f
f
f
fc
R
I==
+g
H
D
D
R≡
*
s
212 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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265. 3.3 Microscopic Diffusion 213
where f
Iis the physical or conductivity cor-
relation factor and gis a two-particle corre-
lation factor (see Sec. 3.3.1.7). For specific
models of interacting particles f
I≤1 (see
Murch and Dyre (1989) for a detailed re-
view). At the limits of an almost full or
empty lattice of charge carriers, f
IÆ1 and
H
R=f. This is compatible, of course, with
Eq. (3-131) for these conditions.
Accordingly, a measurement of the
Haven ratio (obtained by measuring
sand
D*, preferably in the same sample) can in
some cases give falone and therefore the
mechanism of diffusion can be exposed.
However, the interstitialcy mechanism adds
a minor complication because the “charge”
moves two jump distances whereas the
tracer moves only one (see Fig. 3-6). This
is easily taken care of in the analysis so that
H
Ris now written generally as (see for ex-
ample, Murch (1982d))
(3-135)
where rand r
qare the distances moved by
the tracer and charge respectively. Pro-
vided we focus on cases where f
I=1 (either
almost empty or full lattice of charge car-
riers), then H
Rstill has a unique value for
each mechanism. The classic example of
the value of H
Rin identifying the mecha-
nism is Ag motion in AgBr (Friauf, 1957).
It was found that H
Rvaried from 0.5 at low
temperature to 0.65 at high temperature.
Frenkel defects (see Sec. 3.4.4) are ex-
pected and therefore a temperature depen-
dence of H
R. But H
Ris not consistent with
a combination of vacancy and direct inter-
stitial jumps. This would require that H
R
varies from 0.78 to unity. The lower values
of H
Robtained require a contribution from
interstitialcy jumps. It was found that both
collinear and noncollinear interstitialcy
jumps were required to fit the behavior of
H
R.
H
f
f
r
r
q
R
I
=
⎛
⎝
⎜
⎞
⎠
⎟
2
Much less satisfactorily interpreted are
fast ion conductors where the defect con-
centration is high and f
Imust be included
in the analysis of H
R. A review of H
Rfor
such materials has been provided by Murch
(1982d). Further comments on the subject
can be found in the review on correlation
effects in ionic conductivity by Murch and
Dyre (1989). An introduction to the subject
may be found in the book by Philibert
(1991).
3.3.3 The Isotope Effect in Diffusion
The isotope effect, sometimes called the
mass effect, is of considerable importance
in diffusion because it provides one of the
two experimental means of gaining access
to the tracer correlation factor, f(the other
is the Haven ratio; see Sec. 3.3.2). Since f
depends on mechanism (see Sec. 3.3.1.3), a
measurement of the isotope effect can, in
principle, throw light on the diffusion
mechanism operating.
The measurement of the isotope effect
depends on accurately measuring a small
difference between the diffusion coeffi-
cients of two tracers
aand b(Peterson,
1975). This small difference can be related
to fin the following way. For isotropic
tracer and impurity diffusion, the diffusion
coefficients of
aand bcan be written as
D
a= Aw
af
a (3-136)
D
b= Aw
bf
b (3-137)
where Acontains geometrical terms and
defect concentrations which do not depend
on the atom/defect exchange frequency
w.
The correlation factors f
aand f
bhave the
mathematical form of impuritycorrelation
factors because the tracers (i.e. isotopes) in
real experiments are formally impurities in
the sense that their jump frequencies differ
from the host atoms. Most impurity corre-
lation factors for impurity diffusion in oth-www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

266. erwise pure and nearly perfect crystals
have the mathematical form
f
a= u/(w
a+ u) (3-138)
f
b= u/(w
b+ u) (3-139)
where ucontains exchange frequencies
other than the tracer. After taking the ratio
of Eqs. (3-136) and (3-137) and eliminat-
ing uand f
b(say) from Eqs. (3-136), (3-
138), and (3-139), we find that
(D
a– D
b)/D
b=f[(w
a–w
b)/w
b] (3-140)
At this point, the formal distinction be-
tween f
a, f
band the tracer correlation fac-
tor f, the latter referring to a hypothetical
tracer with the same jump frequency as the
host, can be dropped since they are very
similar in numerical value. Knowledge of
the experimental jump frequencies,
w
aand
w
b, is usually not available, of course. Ac-
cordingly, they are expressed in terms of
quantities which are known, such as the
masses of the tracers.
Application of classical statistical me-
chanics to describe the dynamics of the
jumping process leads to the result
(3-141)
where m
aand m
bare the masses of the
tracers
aand b, DKis the fraction of the
total kinetic energy, associated with the
whole motion in the jump direction at the
saddle point of a jump, that actually be-
longs to the diffusing atom. The conversion
of jump frequencies to masses has unfortu-
nately led to the introduction of this new
quantity DK, about which detailed informa-
tion is difficult to obtain. If the remainder
of the lattice is not involved in the jump,
DK=1 (this is the upper limit for DK).
More usually, there is a certain amount of
coupling between the diffusing atom and
the remainder of the lattice and DK<1.
()/
/
www
a bb
b
a−
⎛
⎝
⎜
⎞
⎠
⎟−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=DK
m
m
12
1
From Eqs. (3-140) and (3-141), we find
that
(3-142)
Eq. (3-142) refers to a process where only
one atom jumps. The more general case of
natoms jumping simultaneously, e.g., the
interstitialcy mechanism (where n= 2) (see
Fig. 3-6), is described by
(3-143)
where m
0is the average mass of the non-
tracers. It is common to refer to fDKin
Eqs. (3-142), (3-143) as the “isotope ef-
fect” E(n). Because fand DKare ≤1 then
this also applies to E(n). Accordingly, al-
though a measurement of the isotope effect
may not uniquely determine the mecha-
nism, it can be invaluable in showing
which mechanisms (through their values of
fand n) are not consistent.
There are some mechanisms for which
the appropriate correlation factor does not
have the simple form of Eq. (3-136), for
example a mechanism which has several
jump frequencies. Nonetheless, it is always
possible to derive equations equivalent
to Eqs. (3-142), (3-143) for such cases (see
the review by Le Claire (1970) for details).
In the discussion above, there has per-
haps been the implication that the theory
described here applies only to tracer diffu-
sion in “simple” materials, e.g., pure met-
als, and alkali metal halides. Certainly the
theory was originally developed with these
cases in mind, but it can be used, with cau-
tion, in other more complex situations. One
of the earliest of these was application to
impurity diffusion. In this case the tracers
aand bnow refer to isotopes of an impur-
ity which is chemically different from the
host. When the mechanism is not in any
fK
DDD
mnmmnm
D
=
[( )/ ]
[( (–) )/( (–) )] –
/
a bb
b a−
++111
00
12
fK
DDD
mm
D=
()/
(/)
/
a bb
ba−
−
12
1
214 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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267. 3.3 Microscopic Diffusion 215
doubt, and frequently it is not to the experi-
enced researcher in the field, a measure-
ment of the isotope effect for self-diffusion
yields DK. If the impurity is substitutional,
and provided that DKis assumed to be the
same as for self-diffusion, then an isotope
measurement for the impurity can give the
correlation factor for impurity diffusion.
Since the impurity correlation factor is a
function of local jump frequencies, e.g., the
five-frequency model in the f.c.c. lattice
(see Fig. 3-11), then a knowledge of fcom-
bined with other knowledge of the same
jump frequencies can lead to a knowledge
of some of their ratios (see, for example,
Rothman and Peterson (1967), Bocquet
(1972), and Chen and Peterson (1972,
1973)).
The “impurity form” of the tracer corre-
lation factor is also implied for Manning’s
theory for vacancy diffusion in the random
alloy model (see Sec. 3.3.1.5). Therefore,
isotope effect measurements in concen-
trated alloys which are reasonably well
described thermodynamically by the ran-
dom alloy model can be interpreted along
traditional lines. Similarly, it has been
shown by Monte Carlo computer simula-
tion that the impurity form of fis also valid
for vacancy diffusion in a lattice containing
a high concentration of randomly distrib-
uted vacancies, up to 50% in fact (Murch,
1984b).
When order is increased among the com-
ponents, the impurity form for feven for
vacancy diffusion is increasingly not fol-
lowed. For example, in the stoichiometric
binary alloy AB with order, the correlation
factor, forced to follow the impurity form,
diverges from the actual fbelow the or-
der/disorder temperature (Zhang et al.,
1989b) (see Fig. 3-21). A very similar situ-
ation arises for the case of ordered atoms
diffusing by the vacancy mechanism on a
highly defective lattice such as is often en-
countered at low temperature in fast ion
conductors (Zhang and Murch, 1997). The
implication of these and other findings is
that the isotope effect measured in ordered
materials does not contain the usual tracer
correlation factor but some other correla-
tion factor defined onlyby way of the im-
purity form, Eq. (3-136). This does not
mean that isotope effect measurements in
ordered materials are intrinsically without
meaning, but simply that they cannot be
easily interpreted.
The actual measurement of the isotope
effect in diffusion requires the very accu-
rate measurement of (D
a–D
b)/D
b. This
quantity typically lies between 0.0 and
0.05. Because of the inaccuracies in mea-
surements of D it is not feasible to measure
D
aand D
bin separate experiments. Nor-
mally the isotopes
aand bare co-de-
posited in a very thin layer and permitted
to diffuse simultaneously into a thick sam-
ple. The geometry of the experiment per-
mits Eq. (3-8) to be used for each isotope.
Figure 3-21.Arrhenius plot of Monte Carlo results
for the tracer correlation factor in the simple cubic
lattice. ≤represents the actual tracer correlation fac-
tor;
represents the tracer correlation factor forced
to follow the impurity form; after Zhang et al.
(1989b).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

268. With a little manipulation it is soon found
that
(3-144)
This means that by plotting ln (C
a/C
b) vs.
lnC
a, (D
a–D
b)/D
bcan in fact be obtained
very accurately, avoiding errors due to time
and temperature of the diffusion anneal and
errors in sectioning. The separation of the
isotopes is normally achieved by half-life,
energy spectroscopy, or the use of different
kinds of radiation. Details of how to carry
out careful experiments in this area can be
found in the very comprehensive review of
experimental techniques in diffusion by
Rothman (1984). A more detailed review
of the isotope effect in diffusion has been
provided by Peterson (1975); see also Le
Claire (1970).
3.3.4 The Jump Frequency
In Sec. 3.3.1.2 we showed that a tracer
diffusion coefficient is, in essence, a prod-
uct of an uncorrelated part containing
the jump frequency and a correlated part
containing the correlation factor, see Eq.
(3-76). The Arrhenius temperature depen-
dence of the diffusion coefficient arises
largely from the jump frequency, although
some temperature dependence of the corre-
lation factor in some situations, e.g., al-
loys, cannot be ignored (see Secs. 3.3.1.4
to 3.3.1.6). In this section we will study the
make-up of the atomic jump frequency.
First, for an atom to jump from one site
to another, the defect necessary to provide
the means for the jump must be available.
For the interstitial diffusion mechanism
there is essentially always a vacancy
(really a vacant interstice) available except
at high interstitial concentrations. For other
mechanisms, however, the atom has to
ln ln [( )/ ]
C
C
CDDD
a
b
aa bb
⎛
⎝
⎜
⎞
⎠
⎟
−−= const.
“wait” until a defect arrives. The probabil-
ity of a defect, say a vacancy, arriving at a
particular neighboring site to a given atom
is simply the fractional vacancy concentra-
tion c
v. However, in certain situations such
as alloys showing order, or fast ion conduc-
tors showing relatively high defect order-
ing, the availability of vacancies to the
atoms can be enhanced or depressed from
the random mixing value which c
vsigni-
fies. This topic is discussed further in Secs.
3.4.2 and 3.4.3.
The probability of a given atom being
next to a vacancy is thus gc
vwhere gis the
coordination number. The jump frequency
Gcan we decomposed to
G= gc
vw (3-145)
where
wis usually called the “exchange”
frequency to signify the exchange between
an atom and a neighboring vacant site and
to distinguish it from the actual jump fre-
quency
G. It is permissible at low defect
concentrations to call
wthe defect jump
frequency, since the defect does not have to
“wait” for an atom.
In, say, f.c.c. crystals where the jump
distance ris given by a/÷
–
2, where ais the
lattice parameter and g=12, the tracer dif-
fusion coefficient from Eq. (3-76) can now
be written as
D* = 12c
vw(a
2
/2)f/6 = c
vwa
2
f(3-146)
The same equation is also valid for b.c.c.
crystals. Other examples have been given
by Le Claire (1975).
In many solids, such as metals, alloys
and ionic crystals, c
vis said to be “intrin-
sic” with an Arrhenius temperature depen-
dence. The vacancy formation enthalpy is
contained in the measured activation en-
thalpy for the tracer diffusion process (see
Eq. (3-149)). In some solids, notably ionic
crystals and certain intermetallic com-
pounds, apart from the inevitable tempera-
216 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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269. 3.3 Microscopic Diffusion 217
ture-dependent intrinsic concentration, the
vacancy concentration can also be manipu-
lated by doping or by changing the degree
of nonstoichiometry. In such cases this va-
cancy concentration is sometimes said to
be “extrinsic”, and this contribution to the
total vacancy concentration can easily
swamp the intrinsic vacancy contribution.
The extrinsic vacancy concentration is in-
dependent of temperature. However, in the
case of a change in stoichiometry, tempera-
ture independence of c
vrequires adjust-
ment of the external partial pressure (see
Sec. 3.4.4).
3.3.4.1 The Exchange Frequency
With the defect immediately available,
the atom can jump when it acquires suffi-
cient thermal energy from the lattice for it
to pass over the energy barrier between its
present site and the neighboring site. The
probability of the atom having this thermal
energy is given by the Boltzmann probabil-
ity exp (–G
m
/kT), where G
m
(a Gibbs en-
ergy) is the barrier height, kis the Boltz-
mann constant, and Tis the absolute tem-
perature. The attempt frequency, i.e., the
number of times per second the atom on its
site is moving in the direction of the neigh-
boring site, is the mean vibrational fre-
quency
n
–
. Accordingly, the “jump rate”,
i.e., number of jumps per second,
w, is
given by
w=n
–
exp (–G
m
/kT) (3-147)
Because G
m
is a Gibbs energy, the ex-
change frequency can be partitioned as
w=n
–
exp (S
m
/k)exp(–H
m
/kT) (3-148)
where S
m
is the entropy of migration and
H
m
is the enthalpy of migration. H
m
is
sometimes loosely called the “activation”
enthalpy, but this can lead to misunder-
standings, as we shall now see.
In Sec. 3.3.4.2 we show that the defect
concentration in many cases depends on
temperature in an Arrhenius fashion (Eq.
(3-159)). In such cases, the atomic jump
frequency
G, being the product of w, the
defect concentration, and the coordination
number, is written as from Eq. (3-145)
G= gn
–
exp [(S
m
+ S
v
f)/k]
¥exp [– (H
m
+ H
v
f)/kT] (3-149)
The sum of H
m
and H
f
is more accurately
called the activation enthalpy. We empha-
size again that the activation enthalpy
measured in a diffusion experiment can, for
some solids such as ordered alloys, contain
other contributions such as from correla-
tion effects or defect availability terms.
The enthalpy of migration and the en-
thalpy of defect formation are now rou-
tinely calculated by Lattice or Molecular
Statics methods; see Mishin (1997) and
references therein for examples of such
calculations in grain boundaries.
Although it is straightforward to couch
the argument above concerning
win statis-
tical mechanical terms, this would still be
inadequate because we have neglected the
participation of the remainder of the lattice,
especially the neighbors of the jumping
atom, in the jump process. A more satisfac-
tory treatment has been provided by Vine-
yard (1957). This puts
n
–
on a sounder basis
and gives physical significance to the en-
tropy of migration S
m
.
Vineyard (1957) took a classical statisti-
cal mechanical approach by considering
the phase space of the system comprising
all of its Natoms and one vacancy. Each
point in this phase space represents an
atomic configuration of the system. All
possible configurations are represented.
We are concerned with two neighboring
energy minima in this phase space. These
minima are centered on two points, P and
Q. Point P is the phase space point repre-www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

270. senting an atom adjacent to a vacancy; all
atoms are in their equilibrium positions.
Point Q is the corresponding point afterthe
atom–vacancy exchange. Between the
points P and Q there is a potential energy
barrier. In order to jump, the atom must
pass over this barrier at its lowest point, a
saddle point.
Vineyard (1957) was able to calculate
the jump rate, i.e., the number of crossings
per unit time at the “harmonic approxima-
tion”. This entailed expanding the potential
energy at point P and also at the saddle
point in a Taylor series to second order to
give for
w
(3-150)
where the
n
iare normal frequencies for vi-
bration of the system at point P,
n
i¢are the
normal frequencies at the saddle point and
Uis the difference in potential energy
between the saddle point and the equilib-
rium configuration point, P.
It is possible to transform Eq. (3-150)
into the form of Eq. (3-148) by writing
(3-151)
where the entropy of migration can be
identified with
(3-152)
and
(3-153)
n
n
n
=
=
=
i
N
i
i
N
i
1
3
2
3
0
∏
∏
Sk
i
N
ii
m
=
=
2
3
0
∑ ′ln ( / )nn
i
N
i
i
N
i
SkT
=
=
m
=
1
1
1
∏
∏
−
′
n
n
n
exp ( / )
w
n
n=
=
=
i
N
i
i
N
i
UkT
1
1
∏
∏
′
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
−exp ( / )
where
n
i
0is the frequency of the ith normal
mode with the system constrained to move
normal to the direction joining point P to
the saddle point.
There have been many further considera-
tions of the detailed dynamics of the jump
process, the most notable of these being the
work carried out by Jacucci, Flynn and co-
workers (see reviews by Jacucci (1984)
and Pontikis (1990)).
For general purposes it is often sufficient
to focus on Eq. (3-148) and to note that the
Debye frequency is an adequate represen-
tation of the mean vibration frequency
n
–
.
3.3.4.2 Vacancy Concentration
The change in Helmholtz energy F
f
V
as-
sociated with the formation of one vacancy
is given by
F
f
V
= E
f
V
– TS
f
V
(3-154)
where E
f
V
and S
f
V
are the energy and en-
tropy of formation, respectively. The for-
mation process itself is conceived to be the
removal of an atom from the interior of the
crystal to the surface. The entropy part has
two contributions: a vibrational or thermal
part S
f
vib
arising from the fact that atoms
close to the vacancy have a different vibra-
tional frequency from those far from the
vacancy, and a configurational part S
f
config
which is usually thought to be an ideal mix-
ing entropy, at least for pure metals – but it
is rather more complicated for alloys. The
configurational part for pure metals is eas-
ily found from elementary classical statisti-
cal mechanics (see, for example, Peterson
(1978)). The number of different ways
W
of putting nvacancies and Natoms on N+n
sites assuming indistinguishability within
each group is
(3-155)
W=
()!
!!
Nn
Nn
+
218 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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271. 3.4 Diffusion in Materials 219
The configurational entropy is given by
(3-156)
which, with Stirling’s approximation lnN!
=NlnN–Nfor large N, results in
(3-157)
The total change in free energy when nva-
cancies are produced is
nF
f
V
= nE
f
V
– T(nS
f
config
) (3-158)
After substitution of Eq. (3-157) and put-
ting the derivative ∂ F/∂n= 0 to obtain the
equilibrium number of vacancies, we soon
find that
(3-159)
where c
vis the site fraction of vacancies.
Note that the configurational term has dis-
appeared. The vibrational contribution S
f
vib
is considered to be small and the leading
term in Eq. (3-159) is usually ignored, and
c
v= exp (–E
f
V
/kT) (3-160)
For low pressures we can also assume (not
always safely) that the energy of formation
E
f
V
can be approximated by the enthalpy of
formation H
f
V
.
Divacancies are handled in a similar way
(see, for example, Peterson (1978)). Fren-
kel and Schottky defects in ionic crystals
(see Sec. 3.4.4) can also be analysed along
similar lines (Kofstad, 1972). Alloys, how-
ever, present a special problem because of
uncertainties about reference states includ-
ing the typical surface site where an atom
is to be symbolically placed (see, for exam-
ple, Lim et al. (1990) and Foiles and Daw
(1987)).
Calculations of defect formation ener-
gies are frequently handled by lattice relax-
n
Nn
cSkEkT
+
−==
v vib
f
V
fexp( / ) exp( / )
SkN
N
Nn
n
n
Nn
config
f=−
+
⎡
⎣
⎢
⎤
⎦
⎥
+
+
⎡
⎣
⎢
⎤
⎦
⎥
⎛
⎝
⎜
⎞
⎠
⎟ln ln
Sk k
Nn
Nn
config
f==ln ln
()!
!!W
+⎡
⎣
⎢
⎤
⎦
⎥
ation techniques with computer codes such
as DEVIL and CASCADE (these tech-
niques are described in the book edited by
Catlow and Mackrodt, 1982) and lattice
dynamics and Monte Carlo methods (see,
for example, Jacucci, 1984).
Vacancy concentrations in metals are con-
ventionally measured by quenching, dila-
tometry, and positron annihilation. An intro-
duction to these techniques has been pro-
vided by Borg and Dienes (1988), and a de-
tailed review is also available (Siegel, 1978).
3.4 Diffusion in Materials
Probably every type of solid has, at one
time, been investigated for its diffusion be-
havior. It is impossible in the space here to
cover even the major findings. In this sec-
tion we propose to discuss, at an introduc-
tory level, diffusion in metals and alloys,
and, as an example of ionic crystals,
oxides. Our emphasis here is on the usual
theoretical framework for a description of
diffusion in these materials rather than on
data compilations or reviews. Diffusion
data compilations for metals and alloys can
be found in the Smithells Metals Reference
Book (Brandes, 1983) and in the extensive
compilations in Landolt-Börnstein (Meh-
rer, 1990). The older tracer diffusion data
up to 1970 on metals and oxides has been
collected by Askill (1970) and data on
oxides up to 1970 on metals and oxides
has been collected by Askill (1970) and
data on oxides up to 1980 by Freer (1980).
Extensive compilations of diffusion data
in nonmetallic solids have been published
in Landolt–Börnstein (Beke, 1998 1999).
The journal Defect and Diffusion Form,
formerly Diffusion and Defect Data, regu-
larly publishes abstracts of all diffusion-re-
lated papers and extensive indices are reg-
ularly provided.www.iran-mavad.com
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272. 3.4.1 Diffusion in Metals
3.4.1.1 Self-Diffusion
We have seen (Sec. 3.3.4) that the tracer
diffusion coefficient in solids normally fol-
lows an Arrhenius form. This is usually
written empirically as
D= D
0
exp (–Q/RT) (3-161)
where D
0
is called the pre-exponential fac-
tor, or sometimes the “frequency factor”, Q
is the activation energy for diffusion, Ris
the ideal gas constant, and Tis the absolute
temperature. In most pure metals self-dif-
fusion is characterized by a pre-exponen-
tial D
0
which falls in the range 10
–3
to
5¥10
–6
m
2
s
–1
. This corresponds to activa-
tion entropies which are positive and of the
order of k , the Boltzmann constant. The ac-
tivation energy Qis given fairly closely
(±≈10%) by Q=34T
M.Pt., where T
M.Pt.is
the melting point. These values lead to a
value of the self-diffusion coefficient at the
melting point of about 10
–12
m
2
s
–1
.
Careful dilatometric and quenching ex-
periments on a large number of f.c.c. met-
als indicate the vacancy as being the defect
responsible for “normal” diffusion in f.c.c.
metals. For vacancy diffusion the tracer
diffusion coefficient is written for cubic
lattices as
D* = gc
vwr
2
f/6 (3-162)
where gis the coordination, c
vis the va-
cancy concentration,
wis the vacancy–
atom exchange frequency, ris the jump
distance, and fis the tracer correlation
factor. The decomposition of c
vinto its
enthalpy/entropy parts is given by Eq.
(3-159). The decomposition of
wis given
in Eq. (3-148). The tracer correlation factor
is discussed in Sec. 3.3.1.3.
For some f.c.c. metals, e.g., silver (Roth-
man et al., 1970; Lam et al., 1973), where
diffusion measurements have been made
over a very wide temperature range, there
is a slight curvature in the Arrhenius plots.
This has been variously attributed to a con-
tribution to diffusion from divacancies (see,
for example, Peterson (1978)), or from va-
cancy double jumps (Jacucci, 1984). Be-
cause of the high activation energy, how-
ever, the latter is really only a candidate for
explaining curvature very close to melting
temperature and not over a wide tempera-
ture range.
There is still controversy over the cause
of the curvature in Ag and other f.c.c. met-
als, with one view favoring a temperature-
dependent activation energy and diffusion
by single vacancies (see the review by
Mundy (1992)), and the other tending to fa-
vor, in part, a contribution from divacan-
cies at high temperatures (Seeger, 1997).
The behavior of self-diffusion in b.c.c.
metals is quite varied. Some b.c.c. metals
such as Cr show linear Arrhenius plots.
The alkali metals show slightly curved
plots. Finally, there is a large group of
“anomalous” metals including
b-Zr, b-Hf,
b-Ti, g-U, and e-Pu, which show a strong
curvature and show values of D
0and Q
both of which are anomalously low. There
has been considerable controversy here
also. Again, the monovacancy mechanism
and a temperature-dependent activation en-
ergy have been strongly supported (Mundy,
1992) but this view alone does not seem to
be entirely consistent with the nature of the
curvature. Other processes such as ring
mechanisms and Frenkel pair formation/re-
combination may be operative (Seeger,
1997).
3.4.1.2 Impurity Diffusion
The appropriate diffusion coefficient for
an impurity A (at infinite dilution) in a host
of B when the vacancy mechanism is oper-
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273. 3.4 Diffusion in Materials 221
ating is (for cubic lattices)
D
A=gp
AVw
Ar
2
f
A/6
= D
0
A
exp (–Q
A/RT) (3-163)
where gis the coordination, p
AVis the va-
cancy availability factor to the impurity,
w
Ais the impurity–vacancy exchange fre-
quency, ris the jump distance, and f
Ais the
impurity correlation factor (for details see
Sec. 3.3.1.4). Again, as for self-diffusion in
metals, in many cases we can discern “nor-
mal” behavior wherein the Arrhenius plots
are linear and D
0
A
and Q
Ado not differ
greatly from the self-diffusion values. As
pointed out by Le Claire (1975, 1978), the
relative values are determined principally
by DQ=Q
A–Q
B, where Q
Bis the self-dif-
fusion activation energy for the host. When
D
A>D
B, the impurity is a fast diffuser and
DQis negative. It is often found that this is
correlated to a situation where the valence
of the impurity is greater than that of the
host. The converse is true when D
A<D
B.
The similarity between impurity and host
diffusivities suggest the vacancy mecha-
nism for both. For this mechanism it can
easily be shown that (see, for example, Le
Claire, 1978)
DQ= (H
A
m– H
B
m) + E
B– Q¢ (3-164)
where H
A
m, H
B
mare the activation enthal-
pies of migration for the impurity and host,
E
Bis the vacancy–impurity binding en-
thalpy, and Q ¢is the activation enthalpy
arising from the temperature dependence
of f
A(see Eq. (3-90)). This is normally a
fairly small contribution.
There are “anomalous” impurity diffus-
ers which have diffusion coefficients much
higher than the host. Well-known examples
are the diffusion of noble metals in Pb. As
has been discussed in conjunction with the
substitional-vacancy diffusion mechanism,
the impurities probably exist partly as
interstitials and partly as substitutionals
(see Sec. 3.3.1.1). For a full discussion of
all aspects of impurity diffusion we refer to
Le Claire (1978).
3.4.2 Diffusion in Dilute Alloys
3.4.2.1 Substitutional Alloys
Alloys containing something less than
about 1–2% solute concentrations are con-
sidered to be sufficiently dilute that the dif-
fusion of solute atoms can be considered in
terms of isolated atoms or isolated group-
ings of atoms such as pairs. For many bi-
nary dilute alloys, measurements of the
solute diffusion coefficient D
2(measured
as a tracer diffusion coefficient) have been
made as a function of solute concentration
(we use the subscript 2 to denote the solute
and 0 to denote the solvent). It is usual to
represent the solute diffusion coefficient
empirically as
D
2(c
2) = D
2(0) (1+B
1c
2+B
2c
2
2…) (3-165)
where D
2(0) is the solute diffusion coeffi-
cient for c
2Æ0, c
2is the atomic fraction of
solute and B
1, B
2, … are termed solute en-
hancement factors. D
2(0) is also called the
impurity diffusion coefficient (at infinite
dilution) depending on the context of the
experiment.
Similarly, solvent diffusion coefficients
D
0, also measured as tracer diffusion coef-
ficients, are represented as
D
0(c
2) = D
0(0) (1+b
1c
2+b
2c
2
2…) (3-166)
and b
1, b
2, … are termed solvent enhance-
ment factors.
Of interest are the solute and solvent en-
hancement factors. Naturally, also of inter-
est is the composition (a) of values of the
solute and solvent diffusion coefficients
and (b) of values of their respective activa-
tion energies.www.iran-mavad.com
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274. In Eq. (3-166) D
2(0) arises from isolated
solute atoms, while the term containing B
1
arises from pairs of solute atoms which are
sufficiently close that the solute jump fre-
quency differs from the isolated solute–va-
cancy exchange frequency
w
2. B
2arises
from triplets of solute atoms.
The solute enhancement factor B
1has
been calculated for the f.c.c. lattice by ex-
tending the five-frequency model (see Sec.
3.3.1.4) to include three new frequencies.
These frequencies describe solute jumps
which create, i.e., associate, a new solute
pair (
w
23), dissociate a solute pair (w
24),
and reorient an existing pair (
w
21). Pairs of
solute atoms do not occur randomly when
solute atoms interact. When an interaction
energy E
22between solute atoms is defined
it is straightforward to show that
w
24exp (–E
22/kT) = w
23exp (–E
2B/kT)
where E
2Bis the impurity–vacancy bind-
ing energy. If it is assumed that the impur-
ity correlation factor does not depend on
solute concentration (Stark, 1972, 1974),
then B
1is given by (Bocquet, 1972; Le
Claire, 1978)
If it is assumed that E
22= 0 and that
w
21=w
23=w
2then B
1is reduced to
B
1= 18 [exp (–E
2B/kT) – 1] (3-167a)
There have been no calculations along sim-
ilar lines of B
2.
The solvent enhancement factor b
1can
be calculated by noting the number of jump
frequencies close to the solute. For in-
stance, in the five-frequency model for the
f.c.c. lattice we need to count the number
of solvent frequencies which differ from
BE kT
EkT
EkT
12 2
21
2
22
23
2 612
414
= (3-167)
2B
−+ −
−−
×−−
⎡
⎣
⎢
⎤
⎦
⎥
[ exp( / )]
exp( / )
exp( / )
w
w
w
w
w
0, the solvent exchange frequency far
from the solute. For the f.c.c. lattice the re-
sult for b
1is (Howard and Manning, 1967)
(3-168)
where
c
1and c
2are termed partial correla-
tion factors and are known functions of the
ratios
w
2/w
1, w
1/w
3and w
4/w
0, and f
0is
the correlation factor in the pure lattice.
Eq. (3-168) can be re-expressed as
(3-169)
thereby showing the effect of an altered va-
cancy concentration near a solute by virtue
of the solute–vacancy binding energy E
2B.
There are similar relations for the b.c.c.
lattice. For the impurity model described
in Sec. 3.3.1.4 (Model I), b
1is given by
(Jones and Le Claire, 1972)
(3-170)
and for Model II (see Sec. 3.3.1.4)
(3-171)
where u
1, n
1(w
2/w¢
3, w
3/w¢
3) and u
2, n
2
(w
2/w
3, w
4/w
0) are called mean partial
correlation factors and are known functions
of the frequencies given.
It is possible to express b
1in terms of the
ratioof the solute and solvent diffusion co-
efficients. This aids in the discussion of the
numerical values taken by the enhance-
ment factors. For the f.c.c. lattice, b
1is
given by Le Claire (1978).
b
u
f
f
EkT
1
2
3
0
220 6
14
=
2B
−+
+− exp( / )
w
w
n
b
u
f
f
EkT
1
1
3
0
120 14
6
=
2B
−+
+− exp( / )
w
w
n
b
ff
EkT
1
11
0
2
0
3
018 4 14=
2B
−+ +
⎡
⎣
⎢
⎤
⎦
⎥
×−
cw
w
cw
w
exp( / )
b
ff
1
4
0
1
0
1
3
2
018 4 14=−+ +
⎡
⎣
⎢
⎤
⎦
⎥
w
w
cw
w
c
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275. 3.4 Diffusion in Materials 223
(3-172)
where Fhas been given in Sec. 3.3.1.4 (see
Eq. (3-80)) and f
0is the tracer correlation
factor in the pure lattice. The term in brack-
ets is roughly unity and the impurity corre-
lation factor f
2is about 0.5. The greater the
solute diffusion coefficient is compared
with the solvent diffusion coefficient, then
the more likely it is that b
1is greater than
zero. On the other hand, for slow solute
diffusers b
1will probably be negative but
not less then –18. A few examples taken
from Le Claire (1978) are given in Table
3-2.
The solvent enhancement factor b
2re-
sults from the change in the solvent jump
frequencies in the vicinity of pairsof
solute atoms. Expressions for b
2for the
f.c.c. lattice have been given by Bocquet
(1972).
As noted by Le Claire (1978), the ex-
pressions for b
1and B
1are similar in form.
Provided that the frequency ratios in one
equation do not greatly differ from the
other, as might be expected from relatively
weak perturbations caused by the solute,
then b
1and B
1tend to have the same sign
and roughly comparable magnitudes, apart
from the exp (–E
22/kT) term in the expres-
b
f
f
D
Df F
1
0
2
2
0
113 2
01318
4
1
41 4
414
=−+
−
+
+
⎡
⎣
⎢
⎤
⎦
⎥
cww c
ww(/)
(/) sion for B
1. These comments are borne out
by the data in Table 3-2.
The relative values of the diffusion coef-
ficients of solute and solvent are dictated
largely by the difference in activation ener-
gies for solute and solvent diffusion rather
than by differences in the pre-exponen-
tial factors. The difference DQ=Q
2–Q
0
(where Q
2and Q
0are the activation ener-
gies for solute and solvent respectively) of-
ten seems to be closely related to the differ-
ence in valencies of solute and solvent, Z
2
and Z
0. When DQ is negative (fast solute
diffusion), Z
2>Z
0. On the other hand,
when DQis positive Z
2<Z
0. These matters
are discussed further in a detailed review
by Le Claire (1978). The reader is also di-
rected to a recent commentary (Le Claire,
1992).
An approximate approach which acts as
a transition between the models described
above and the concentrated alloy models of
Sec. 3.4.3 is the complex model initiated
first by Dorn and Mitchell (1966) and de-
veloped by Faupel and Hehenkamp (1986,
1987). When impurities (B) have a positive
excess charge, vacancy–impurity com-
plexes of one vacancy and iimpurity atoms
are likely to form. Assuming that the bind-
ing free energy (–G
Bi) is independent of
the configuration of the impurity atoms,
Dorn and Mitchell (1966) wrote for the
Table 3-2.Some values of solute and solvent enhancement factors, from Le Claire (1978).
Alloy system Lattice D
2/D
0 b
1 B
1 T(°C)
(Solvent–solute)
Ag–Pd f.c.c. 0.04 – 8.2 – 7.5 730
Ag–Au f.c.c. 3.7 7.0 5.5 730
Ag–Cd f.c.c. 5.3 13.0 8.5 730
Ag–In f.c.c. 8.4 37 30 730
Ag–Tl f.c.c. 0.26 – 1.2 – 0.56 730
b-Ti–V b.c.c. 1.26 – 4.3 – 2.5 1400
V–Ti b.c.c. 1.66 16 27 1400
a-Fe–Si b.c.c. 1.64 20.4 12.4 1427www.iran-mavad.com
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276. mole fraction of vacancies in the ith com-
plex c
vi
(3-173)
where c
v(0) is the mole fraction of vacan-
cies in the pure metal (A) and c
0and c
2are
the mole fractions of solvent (A) and solute
(B) respectively, and gis the coordination.
Bérces and Kovács (1983) modified Eq. (3-
173) to take into account possible configu-
rations of the complexes.
Within this complex model framework
Faupel and Hehenkamp (1986, 1987) have
introduced averageeffective jump fre-
quencies per solvent (A) atom ·
w
A
eff
Ò
iand
average effective jump frequencies per
solute (B) atom ·
w
B
eff
Ò
iin the ith complex
(the latter specifically includes correlation
effects – which are always important for
the solute). The normalized solvent diffu-
sion coefficient can now be written as
(3-174)
Similarly, the normalized solute diffusion
coefficient can be written as
These equations for the normalized solute
and solvent diffusion coefficients can be
expanded in terms of c
2to give explicit ex-
pressions for the various solute and solvent
enhancement factors by way of comparison
with Eqs. (3-165) and (3-166); see Le
Claire (1992).
It has been found, in a number of careful
studies in f.c.c. systems (Ag–Sb, Ag–Sn)
where the impurities have excess positive
charge, that solvent diffusion on alloying is
Dc
D
c
i
f i
cc G G kT
i
i
i
i
02
0
0
1
2 22
0
1
2
1
10
()
()
exp[( )/ ]
= (3-175)
=
eff
B
BB
g
g
gg
g
−
−−
−
〈〉 ⎛
⎝
⎜
⎞
⎠
⎟
×−∑
w
w
Dc
D
c
f i
cc GkT
i
i
ii
i
02
0
0
1
1
1
00
0
1
20
1()
()
exp( / )
=
=
eff
A
B
g
g
g
g
−
−
−−
−
〈〉 −⎛
⎝
⎜
⎞
⎠
⎟
×∑
w
w
cc cc GkT
i
ii
ivv B= ( ) exp( / )0
0
02
g
g⎛
⎝
⎜
⎞
⎠
⎟ −
enhanced in a nonlinear way (Hehenkamp
et al., 1980; Hehenkamp and Faupel, 1983).
Moreover, independent measurements of
the vacancy concentration on alloying have
shown a lineardependence between nor-
malized vacancy concentration and the
normalized solvent diffusion coefficient
(Hehenkamp et al., 1980; Hehenkamp and
Faupel, 1983). The nonlinear enhancement
of solvent diffusion upon alloying is thus
apparently due to a corresponding nonlin-
ear increase in the vacancy concentration
on alloying. These findings are very nicely
described by the complex model.
Despite its obvious limitations, the com-
plex model can be useful quantitatively up
to 5 at.% of solute and qualitatively even
higher; see Le Claire (1992).
3.4.2.2 Interstitial Alloys
Elements such as H, N, O, and C dis-
solve interstitially in metals and diffuse by
the interstitial mechanism. The diffusion
coefficients here are often measured by re-
laxation techniques or outgassing. Tracer
diffusion is difficult to measure in the case
of N and O because of the lack of suitable
radioisotope; however, other techniques
are available (see Sec. 3.5.1).
For interstitials at infinite dilution, the
interstitials move independently (by the
interstitial mechanism, see Fig. 3-5). The
vacancy concentration (vacant interstice) is
unity, the correlation factor is unity (a com-
plete random walk), and the tracer diffu-
sion coefficient is given simply by (from
Eqs. (3-76) and (3-145))
D* = g
w
0r
2
/6 (3-176)
where
w
0is the interstitial/vacant interstice
“exchange” frequency.
In a number of cases the interstitial con-
centration can become sufficiently high for
the interstitials to interfere with one an-
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277. 3.4 Diffusion in Materials 225
other and this interference may have an im-
portant effect on diffusion. An example is
C in austenite where the carbon interstitials
can fill roughly 8% of the octahedral voids.
The two theoretical approaches taken to
cope with the problem are reminiscent of
the situation found in the modeling of sub-
stitutional alloys (see Sec. 3.4.2.1). In the
first, the exchange frequencies for the so-
lute are specified. In the second, the ex-
change frequencies are specified indirectly
by way of interaction energies.
For the first approach, McKee (1980a,
b) and Le Claire (1981) conceived a four-
frequency model for interstitial solute dif-
fusion in the f.c.c. lattice (the octahedral
voids of the f.c.c. lattice). Two distinct spe-
cies are considered: the isolated interstitial
with a jump frequency of k
0, and paired
interstitials, which rotate with frequency
u
1, dissociate with frequency u
3, and asso-
ciate with frequency u
4. The following ex-
pression for the tracer diffusion coefficient
of the interstitial solute can then be derived
(Le Claire (1981) using Howard’s (1966)
random walk method)
where c
pis the site fraction concentration
of paired interstitials and cis the site frac-
tion of interstitials. Using the pair associa-
tion method, McKee (1980a, b) derived a
similar result valid for weak binding of the
paired interstitials. Explicit expressions for
the tracer correlation factor can be found in
the original papers.
It was also possible to determine an ex-
pression for the chemical diffusion coeffi-
cient, D˜, relating to diffusion of the solute
in its own concentration gradient. In the
weak binding limit the expression for D˜is
Drk
c
c
uu
uu
uu
kk
u
u
*
()
()
= (3-177)
p
4
6
414
23
27
12 7
2
013
13
2
13
00
3
4
+
⎡
⎣
⎢
+
⎛
⎝
⎜
−
−
+
−−
⎞
⎠
⎟
⎤
⎦
⎥
⎥
(McKee, 1981)
D˜=–
1
3
r
2
[12k
0+ 12u
4/u
3]
¥[4u
1+7u
3–12k
0+7(u
4–k
0)u
3/u
4]
¥c{1 +c[1 + 12 (1 –u
4/u
3)]} (3-178)
where ris the jump distance. Eq. (3-178)
containsa thermodynamic factor.
The above model, where the frequencies
are specified explicitly, is probably only
valid for 1–2% of interstices occupied. At
higher concentrations many more frequen-
cies are required with the result that this
approach becomes unwieldy. For the sec-
ond approach a lattice gas model is used to
represent the solute interstitials and their
lattice of interstices. High concentrations
of interstitials can in principle be handled
with this model. In this rigidmodel the at-
oms are localized at sites and pairwise
interactions among atoms are specified.
The exchange frequency of a solute atom to
a neighboring site can be written as (Sato
and Kikuchi, 1971)
w=nexp(g
nnf
nn/kT) exp(–U/kT) (3-179)
although many other choices are possible.
In Eq. (3-179)
nis the vibration frequency,
g
nnis the number of occupied interstices
that are nearest neighbors to a solute,
f
nnis
the solute–solute “binding” energy (posi-
tive or negative), and Uis the migration en-
ergy for an isolatedsolute atom. In effect
the solute neighbors can assist or impede
the diffusion process depending on the sign
of
f
nn.
By its nature this model requires a statis-
tical mechanical approach for its analysis.
One method used is the PPM (Sato and
Kikuchi, 1971; Sato, 1989, see also Sec.
3.3.1.5); another is the Monte Carlo com-
puter simulation method (Murch, 1984a).
It is convenient to decompose the tracer
diffusion coefficient into
D* = gr
2
VWfexp (–U/kT)/6 (3-180)www.iran-mavad.com
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278. where Vis the vacancy availability factor
(the average availability of neighboring va-
cant interstices to an interstitial solute
atom, and is a more general quantity than
c
v), and W represents the effect of the envi-
ronment on the jump frequency. It is in ef-
fect the statistical average of the first term
in Eq. (3-179). The same lattice gas model
has also been applied extensively to fast
ion conductors (see, for example, Murch
(1984a)).
Chemical diffusion can readily be ex-
pressed along similar lines (Murch, 1982c)
as
D˜=gr
2
VWnf
I (3-181)
¥exp (–U/kT) (∂
m/kT/∂lnc)/6
where
mis the chemical potential of the
solute and f
Iis the physical correlation fac-
tor (see Sec. 3.3.1.7). The physics for f
I
are not contained in the first approach (Eq.
(3-178)).
These approaches seem to describe C
tracer and chemical diffusion in
g-Fe fairly
well (McKee, 1980a, b, 1981; Murch and
Thorn, 1979) but other applications have
not been made because of the lack of suit-
able data, especially for D*.
Quantum effects are an important ingre-
dient in the description of H diffusion in
metals; see the reviews by Völkl and Ale-
feld (1975), Fukai and Sugimoto (1985),
and Hempelmann (1984).
3.4.3 Diffusion in Concentrated Binary
Substitional Alloys
Dilute alloy models, see Sec. 3.4.2.1,
cannot be extended very far into the con-
centrated regime without rapidly increas-
ing the number of jump frequencies to an
unworkable level. As a result, models have
been introduced which limit the number of
jump frequencies but only as a result of
some loss of realism. The first of these is
the random alloy model introduced by
Manning (1968, 1970, 1971). The second
is the interacting bond model which has
been extensively developed by Kikuchi
and Sato (1969, 1970, 1972).
In the random alloy model, the atomic
components are assumed to be mixed ran-
domly and the vacancy mechanism is as-
sumed. The atomic jump frequencies
w
A
and w
Bfor the two components A and B
are specified and these do not change with
composition or environment. The tracer
diffusion coefficient of, say, A is given by
D*
A= gw
Ar
2
c
vf
A/6 (3-182)
where gis the coordination, ris the jump
distance, and c
vis the vacancy site fraction
(see Sec. 3.3.4.2). The only theoretical re-
quirement here is to calculate the tracer
correlation factor f
A; this has been de-
scribed in Sec. 3.3.1.5 on correlation ef-
fects in diffusion. The random alloy model
performs very well in describing the diffu-
sion behavior of many alloy systems which
are fairly well disordered. This subject is
discussed extensively in the review by
Bakker (1984) to which we refer. A much
more accurate method for dealing with f
A
has been given by Moleko et al. (1989).
For alloys which exhibit order, Man-
ning’s theory, based on the random alloy
model, still performs reasonably well in
describing certain aspects of diffusion be-
havior, notably correlation effects which
can be expressed in terms of the tracer dif-
fusion coefficients, see Sec. 3.3.1.8 and the
review by Murch and Belova (1998). A
wholly different approach pioneered by
Kikuchi and Sato (1969) is the PPM (see
Sec. 3.3.1.5) in which the tracer diffusion
coefficient is expressed in terms of quan-
tities which are statistically averaged over
the atomic configurations encountered in
the alloy. The tracer diffusion coefficient
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279. 3.4 Diffusion in Materials 227
of, say, A is now written as
D*
A= gw
–
Ar
2
p
–
Avf
–
A/6 (3-183)
where
w
–
Ais an averaged exchange fre-
quency which includes the effect of the en-
vironment on the jump frequency, p
–
Avis the
vacancy availability factor, i.e., the prob-
ability of finding a vacancy next to an A
atom (see below), andf
–
Ais the tracer corre-
lation factor. We have discussed the corre-
lation factorf
–
Ain Sec. 3.3.1.5, to which we
refer. The quantity
w
–
Ais the average of the
exchange frequency for an A atom with a
vacancy in particular configuration:
w
A=n
Aexp (–U
A/kT) (3-184)
¥exp [(g
AE
AA+g
BE
AB)/kT]
where
n
Ais the vibration frequency, U
Ais
the migration energy (referred to some ref-
erence) for a jump, g
Ais the number of A
atoms that are nearest neighbors to a given
A atom (itself next to a vacancy), g
Bis the
number of B atoms which are likewise
nearest neighbors to the A atom, and E
AA
and E
ABare nearest neighbor interactions
(assumed negative here). This equation is
essentially the binary analogue of Eq. (3-
179). We see that the neighbors of the atom
A can increase or decrease the apparent mi-
gration energy.
The use of c
valone in the expression for
D*
Ain Eq. (3-182) implies that either com-
ponent “sees” the vacancy equally. This
obviously cannot generally be true. The in-
troduction of the quantity p
–
Avin Eq. (3-
183) (and p
–
Bv) is a recognition that vacan-
cies are somewhat apportioned between the
two atomic components. This is not a small
effect, particularly in alloys which show
separate sublattices.
The overall activation energy for diffu-
sion is now rather complicated. Apart from
the usual contribution from the vacancy
formation energy and reference migration
energy, we have a complicated contribution
from
w
–
Abecause the configurations change
with temperature. We also have contribu-
tions from p
–
Avand the tracer correlation
factor. In order to make sense of the experi-
mental activation energies, we need to
model the system in question; there seems
to be no alternative. Few detailed applica-
tions of this model have yet been made.
One detailed application has been made to
b-CuZn (Belova and Murch, 1998). Fur-
ther details can be found in the review by
Bakker (1984), see also the review by
Murch and Belova (1998).
Interdiffusion in binary concentrated al-
loys has been dealt with largely in Secs.
3.2.3.2 and 3.3.1.8. Some recent calcula-
tions (which use the “bond model” de-
scribed above) have been performed by
Zhang et al. (1988) and Wang and Akbar
(1993), see also the review by Murch and
Belova (1998).
3.4.4 Diffusion in Ionic Crystals
In this section diffusion in ionic crystals
(exemplified here by oxides) will be briefly
discussed. A more detailed discussion has
been provided by Kofstad (1972) and
Schmalzried (1995). Diffusion in other
ionic crystals, especially alkali metal ha-
lides and silver halides, has been reviewed
by, for example, Fredericks (1975), Laskar
(1990, 1992), Monty (1992).
3.4.4.1 Defects in Ionic Crystals
In order to discuss diffusion in ionic
crystals, we need first to discuss at an ele-
mentary level the types of defects which
arise. Of almost overriding concern in
ionic crystals is the requirement of charge
neutrality.
Let us first consider stoichiometriccrys-
tals, say the oxide MO, and “intrinsic” de-
fect production. The Schottky defect (actu-www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

280. ally a pair of defects) consists of a vacant
anion site and a vacant cation site. It arises
as a result of thermal activation and not
through interaction with the atmosphere.
The Schottky defect generation is written
as a chemical reaction, i.e.,
0
[V≤
M+ V
O

(3-185)
where 0 refers to a perfect crystal, V≤
Mis a
vacant (V) metal (M) site, the primes refer
to effective negative charges (with respect
to the perfect crystal), V
O

is a vacant (V)
oxgen (O) site and the dots refer to effec-
tive positive charges. This is the Kröger-
Vink defect notation (see, for example,
Kofstad (1972)).
We can write an equilibrium constant K
s
for the reaction in the following form, valid
for low defect concentrations:
K
s= [V≤
M] [V
O

] (3-186)
where the brackets [ ] indicate concentra-
tions. Eq. (3-186) is usually called the
Schottky product. K
scan be expressed in
the usual way
K
s= exp (–G
s
f/kT) (3-187)
where G
s
fis the Gibbs energy of formation
of the Schottky defect, which can be parti-
tioned into its enthalpy H
s
fand entropy S
s
f
parts.
The other type of defect occurring in the
stoichiometric ionic crystal is the Frenkel
defect. The Frenkel defect (actually a pair
of defects) consists of a cation interstitial
and cation vacancy or an anion interstitial
and anion vacancy. In the latter case it has
also been called an anti-Frenkel defect, al-
though this nomenclature is now relatively
uncommon. Like the Schottky defect, the
Frenkel defect is thermally activated. The
Frenkel defect generation is also written as
a chemical reaction, for example for cat-
ionic disorder:
M
M[M
i

+ V≤
M (3-188)
where M
Mis a metal atom (M) on a metal
site (M), M
i

is an effectivelydoubly posi-
tively charged metal ion on interstitial isite
and V≤
Mis an effectivelydoubly negatively
charged metal ion vacancy. We have as-
sumed double charges here purely for illus-
trative purposes.
The equilibrium constant for the Frenkel
defect reaction can be written as
K
F= [M
i

] [V≤
M] (3-189)
provided that the defect concentration is
low. This equation is often called the Fren-
kel product. Again, the equilibrium con-
stant K
Fcan be expressed as exp (–G
f
F
/kT)
where G
f
F
is the Gibbs energy of formation
of the Frenkel defect, which again can be
partitioned into its enthalpy H
f
F
and en-
tropy S
f
F
parts.
3.4.4.2 Diffusion Theory in Ionic Crystals
In most mechanisms of diffusion except
the interstitial mechanism, an atom must
“wait” for a defect to arrive at a nearest
neighbor site before a jump is possible (see
Sec. 3.3.4). Thus the jump frequency in-
cludes a defect concentration term, e.g., c
v,
the vacancy concentration. Let us examine
an example for diffusion involving the
Frenkel defect. Although both an intersti-
tial and a vacancy are formed, in oxides
one of them is likely to be much more mo-
bile, i.e., to have a lower migration energy,
than the other. (In the case of stoichiomet-
ric UO
2, for example, theoretical calcula-
tion of migration energies suggests a much
lower migration energy for the oxygen va-
cancy than for the interstitial (by either
interstitial or interstitialcy mechanisms)
(Catlow, 1977).) At the stoichiometric
composition,
[M
i

] = [V≤
M] (3-190)
228 3 Diffusion Kinetics in Solidswww.iran-mavad.com
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281. 3.4 Diffusion in Materials 229
so that the cation vacancy concentration is
given by
[V≤
M] = c
v= exp (–G
f
F
/2kT) (3-191)
= exp(–H
f
F
/2kT) exp(S
f
F
/2k) (3-192)
The measured activation enthalpy for dif-
fusion will be the migration enthalpy plus
halfthe Frenkel defect formation enthalpy.
The reader might well ask how we know
we are dealing with a Frenkel defect and
not a Schottky defect, and if we do know it
is the Frenkel defect then how do we know
it is the vacancy mechanism that is operat-
ing and not the interstitialcy mechanism?
In general, we have to rely on independent
information, principally computer calcula-
tions of defect formation and migration en-
thalpies, but structural information such as
is provided by neutron diffraction and ther-
modynamic information is also useful.
Another process of interest here is the in-
trinsic “ionization” process whereby an
electron is promoted from the valence band
to the conduction band leaving behind a
hole in the valence band. In the Kröger-
Vink notation, we write for the intrinsic
ionization process (the electrons and holes
may be localized on the metal atoms)
0
[e¢+ h

(3-193)
All ionic crystals are capable, in princi-
ple, of becoming nonstoichiometric. The
limit of nonstoichiometry is largely dic-
tated by the ease with which the metal ion
can change its valence and the ability of the
structure to “absorb” defects without re-
verting to some other structure and thereby
changing phase. Nonstoichiometry can be
achieved by either 1) an anion deficiency,
which is accommodated by either anion va-
cancies, e.g., UO
2–x, or metal interstitials,
e.g., Nb
1+yO
2or 2) an anion excess, which
is accommodated by either anion intersti-
tials, e.g., UO
2+xor metal vacancies, e.g.,
Mn
1–yO. As an example we will deal with
an anion excess accommodated by metal
vacancies.
The degree of nonstoichiometry and
therefore the defect concentration accom-
panying it are functions of temperature and
partial pressure of the components. The de-
fects produced in this way are sometimes
said to be “extrinsic”, but this terminology
is probably to be discouraged since they
are still strictly intrinsic to the material. In
an oxide it is usual, at the temperatures of
interest, to consider only the partial pres-
sure of oxygen since it is by far the more
volatile of two components. In carbides,
where the temperatures of diffusion of
interest are much higher, the metal partial
pressure can be comparable to the carbon
partial pressure and either can be manipu-
lated externally.
The simplest chemical reaction generat-
ing nonstoichiometry is
–
1
2
O
2(g) [O
¥
O
+ V
¥
M
(3-194)
where O
¥
O
is a neutral (¥) oxygen ion on an
oxygen site and V
¥
M
is a neutral metal va-
cancy. Physically, this corresponds to oxy-
gen being adsorbed on the surface to form
ions and more lattice sites, thereby effec-
tively making metal vacancies. These va-
cancies diffuse in by cation–vacancy ex-
change until the entire crystal is in equilib-
rium with the atmosphere. Excess oxygen
nonstoichiometry corresponds also to oxi-
dation of the metal ions. Here in this exam-
ple, the holes produced are conceived to be
associated very closely with the metal va-
cancies. The holes can be liberated, in
which case the reaction can now be written
–
1
2
O
2(g) [V¢
M+ O
¥
O
+ h

(3-195)
If the hole is then localized at a metal ion,
we can consider this as the chemical equiv-
alent of having M
3+
ions formally present
in the sublattice of M
2+
ions.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

282. Assuming reaction Eq. (3-194) is the
preferred one, we find that the equilibrium
constant for this reaction is
(3-196)
thereby immediately showing that the
metal vacancy concentration depends on
the partial pressure of oxygen in the fol-
lowing way:
[V
¥
M
] = K
1p
O
2
1/2
(3-197)
Accordingly, the metal vacancy concentra-
tion is directly proportional to p
O
2
1/2
.
Assuming that the reaction in Eq. (3-
195) is the preferred one, we find that the
equilibrium constant for this reaction is
(3-198)
The concentration of [O
¥
O
] is essentially
constant and is usually absorbed into K
2.
The condition of electrical neutrality re-
quires that
[h

] = [V¢
M] (3-199)
thereby showing that the metal vacancy
concentration depends on the partial pres-
sure of oxygenin the following way:
[V¢
M] = K
2
1/2p
2
1/4 (3-200)
Accordingly, the metal vacancy concentra-
tion is directly proportional to p
O
2
1/4
.
Let us assume that the migration of
metal ions will principally be via vacancies
in the nonstoichiometric region, since they
are the predominant defect. Since the tracer
diffusion coefficient of the metal ions de-
pends directly on vacancy concentration
(see Sec. 3.3.4), then the tracer diffusion
coefficient will depend on oxygen partial
pressure in the same way that either [V
¥
M
]
or [V¢
M] does. A measurement of the tracer
diffusion coefficient as a function of oxy-
K
p
2
12
2
=
h][O ][V ]
OM
O[
/
×
′
K
p
1
122
=
[V ]
M
O
×
/
gen partial pressure will expose the actual charge state of the vacancy, i.e., V
¥
M
or V¢
M.
In practice, this kind of differentiation
between charge states and even defect types does not often occur unambiguously. The result of plotting logD* versus logp
O
2
often does not show precisely one slope and often there is also some curvature. These effects can sometimes be associated with contributions to the diffusion from the three types of vacancy, V
¥
M
, V¢
M, and V≤
M,
e.g., Co
1–dO (Dieckmann, 1977). Probably
the most successful and convincing appli- cation of this mass-action law approach has made the defect types in Ni
1–dO compat-
ible with data from tracer diffusion, electri- cal conductivity, chemical diffusion, and deviation from stoichiometry/partial pres- sure (Peterson, 1984).
At higher partial pressures of oxygen,
which in some cases such as Co
1–dO,
Mn
1–dO, and Fe
1–dO can result in large
deviations from stoichiometry, we can log- ically expect that defect interactions could play an important role. First, the concentra- tions in the mass-action equations should be replaced by activities. This so greatly complicates the analysis that progress has been fairly slow. The retention of concen- trations rather than activities may be per- missible, however, because there is some evidence that non-ideal effects tend to can- cel in a logD/logp
O
2
plot (Murch, 1981).
At high defect concentrations, defect clus- tering to form complex defects which themselves might move or act as sources and sinks for more mobile defects, is pos- sible. Two very well-known examples, identified by neutron scattering, are the Koch–Cohen clusters in Fe
1–dO (Koch
and Cohen, 1969) and the Willis cluster in UO
2+x(Willis, 1978). Although attempts
have been made to incorporate such defects into mass-action equations, it is probably fair to say that the number of adjustable pa-
230 3 Diffusion Kinetics in Solidswww.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

283. 3.5 Experimental Methods for Measuring Diffusion Coefficients 231
rameters thereby resulting and the lack of
any precise independent information on the
mobility or lifetime of the defect complexes
make any conclusions reached rather spec-
ulative. New statistical mechanical ap-
proaches have considerable promise, how-
ever (see the review by Murch (1995)).
Somewhat analogous to changing the de-
fect concentration, i.e., deviation from stoi-
chiometry by changing the oxygen partial
pressure, is the doping of ionic crystals
with ions of a different valence from the
host metal ion in order to produce defects.
As an example, consider the solubility of
an oxide
2O
3in an oxide MO
2. The im-
plication in the stoichiometry
2O
3is that
the ions have a valence of +3, compared
with + 4 for M in the oxide MO
2. For
charge neutrality reasons the doped oxide
adopts either oxygen vacancies, metal in-
terstitials, or some electronic defect, the
choice being dictated by energetics. Let us
consider the first of these:

2O
3ÆV
O

+ 2¢
M+ 3O
O (3-201)
where ¢
Mis a ion on a M site. The site
fraction of oxygen vacant sites is not quite
directly proportional to the dopant site
fraction because extra normal oxygen sites
are created in the process. The diffusion
coefficient of oxygen here is proportional
to the concentration of vacant oxygen sites.
For very large deviations from stoichiom-
etry, or high dopant concentrations, those
extrinsic defects greatly outnumber the in-
trinsic defects. However, at low dopant con-
centrations (or small deviations from stoi-
chiometry), all the various mass-action laws
must be combined. This complicates the
analysis, and space prevents us from dealing
with it here. For a more detailed discussion
of this and the foregoing, we refer to Kof-
stad (1972) and Schmalzried (1995).
Generally, cation diffusion in the oxides
of the transition metals (which show non-
stoichiometry (cation vacancies) in the cat-
ion sublattice) is generally well understood
except at large deviations from stoichiome-
try (see the review by Peterson (1984)).
Other oxides such as MgO and Al
2O
3, al-
though apparently simple, are less well
understood because of the low intrinsic de-
fect population and the presence of extrin-
sic defects coming from impurities (Subba-
Rao, 1985). An understanding of oxygen
diffusion in oxides has been hampered by
the lack of suitable radioisotopes, but
methods using
18
O and secondary ion mass
spectrometry (SIMS) have rapidly changed
the situation (Rothman, 1990). Ionic con-
ductivity (where possible) (see Sec. 3.3.2
for the relation to diffusion) has tradition-
ally been the measurement which has given
much more information on oxygen move-
ment; see, for example, the review by
Nowick (1984).
3.5 Experimental Methods for
Measuring Diffusion Coefficients
In this section we briefly discuss the
more frequently used methods of measur-
ing diffusion coefficients.
3.5.1 Tracer Diffusion Methods
By far the most popular and, if per-
formed carefully, the most reliable of the
experimental methods for determining
“self” and impurity diffusion coefficients
is the thin-layer method. Here, a very thin
layer of radiotracer of the diffusant is de-
posited at the surface of the sample. The
deposition can be done by evaporation,
electrochemical methods, decomposition
of a salt, sputtering, etc. The diffusant is
permitted to diffuse for a certain time t at
high temperature, “high” being a relative
term here. If the thickness of the layer de-www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

284. posited is much smaller than ÷
---
Dt, then Eq.
(3-8) describes the time evolution of the
concentration–depth profile. After diffu-
sion, the concentration–depth profile in
the sample is established by sectioning and
counting the radioactivity in each section.
A number of techniques are available for
sectioning. For thick sections ≥3µm, me-
chanical grinding is the standard method.
Microtomes can be used for sections of
about 1 µm, electrochemical methods for
≥5 nm, and sputtering for ≥1 nm. The dif-
fusion coefficient is obtained from the
slope of the lnC(x,t) versus x
2
plot (see
Eq. (3-8)). Fig. 3-22 shows an exemplary
tracer concentration profile of this type
(Mundy et al., 1971).
A variant of the method is the “residual
activity” or “Gruzin (1952) method”.
Rather than counting the activity in each
section, the activity remaining in the sam-
ple is determined. This requires an integra-
tion of Eq. (3-8) and a knowledge of the ra-
diation absorption characteristics of the
material. At the limits of very soft or very
hard radiation the method can give com-
parable accuracy to the counting of sec-
tions.
A few elements do not have convenient
radioisotopes. The diffusion part of the ex-
periment is still performed in much the
same way but now with stable isotopes. In
some cases, e.g.,
18
O, the diffusant source
is in the gas phase, although a thin source
of oxide containing
18
O can be deposited in
some cases. After diffusion, nuclear reac-
tion analysis of
18
O can be used (see, for
example, the review by Lanford et al.
(1984)), in order to establish the concentra-
tion profile. Much more popular recently,
especially for oxygen, is SIMS (see, for ex-
ample, the reviews by Petuskey (1984),
Kilner (1986) and Manning et al. (1996)).
Excellent detailed (and entertaining) ex-
positions and critiques of all methods
available for measuring tracer diffusion co-
efficients in solids have been provided by
Rothman (1984, 1990).
3.5.2 Chemical Diffusion Methods
The chemical or interdiffusion coeffi-
cient can be determined in a variety of
ways. For binary alloys, the traditional
method has been to bond together two sam-
ples of different concentrations. Interdiffu-
sion is then permitted to occur for a time t
and at high temperature. The concentration
profile can be established by sectioning
and chemical analysis, but since about the
late 1960s the use of an electron micro-
probe has been the usual procedure for ob-
taining the concentration profile directly. If
the compositions of the starting samples
are relatively close, and the interdiffusion
coefficient is not highly dependent on com-
position, then analysis of the profile with
Eq. (3-12) can lead directly to the interdif-
fusion coefficient D ˜at the average compo-
sition. The more usual procedure is to use
232 3 Diffusion Kinetics in Solids
Figure 3-22.A tracer concentration profile for self-
diffusion in potassium at 35.5 °C (Mundy et al.,
1971).www.iran-mavad.com
+ s e l 〈'4 , kp e r i 〈&s ! 9 j+ N 0 e

285. 3.5 Experimental Methods for Measuring Diffusion Coefficients 233
the Boltzmann–Matano graphical integra-
tion analysis (see Eqs. (3-13) and (3-14)) to
obtain the interdiffusion coefficient and its
composition dependence. If fine insoluble
wires (to act as a marker) are also incorpo-
rated at the interface of the two samples,
then the marker shift or Kirkendall shift
with respect to the original interface can
be measured. This shift, in association
with the interdiffusion coefficient, can be
used to determine the intrinsic diffusion
coefficients of both alloy components at
the composition of the marker (see Sec.
3.2.2.4).
The rate of absorption or desorption of
material from the sample can also be used
to determine the chemical diffusion coeffi-
cient. This method is useful only where one
component is fairly volatile, such as oxy-
gen in some nonstoichiometric oxides and
hydrogen in metals. The surface composi-
tion is normally assumed to be held con-
stant. The concentration profile can be de-
termined by sectioning or electron micro-
probe analysis and can be analysed with
the aid of Eq. (3-9) to yield the interdiffu-
sion coefficient. The Boltzmann–Matano
analysis can be used if there is a composi-
tion dependence.
The amount of material absorbed or de-
sorbed from the couple, e.g., weight
change, can be used with Eq. (3-11) to
yield the interdiffusion coefficient. Eqs. (3-
9) and (3-11) refer to situations when ÷
---
Dt
is very small compared with the geometry
of the sample, i.e., diffusion into an infinite
sample is assumed. When the sample must
be assumed to be finite, e.g., fine particles,
other, generally more complex, solutions
are required; see Crank (1975). A closely
related experiment can be performed where
the gas phase is chemically in equilibrium
with the solid, but is dosed with a stable
isotope. An example is
18
O, a stable iso-
tope of oxygen. The absorption by the solid
of
18
O (exchanging with
16
O) can be moni-
tored in the gas phase with a mass spec-
trometer in order to lead to the tracerdiffu-
sion coefficient of oxygen (see, for exam-
ple, Auskern and Belle (1961)). A draw-
back of methods which do not rely on sec-
tioning is the necessary assumption of
rapid gas/surface reactions.
3.5.3 Diffusion Coefficients
by Indirect Methods
There are a number of methods for ob-
taining diffusion coefficients that do not
depend on solutions of Fick’s Second Law.
Unfortunately space does not permit a de-
tailed discussion of these. We shall men-
tion them here for completeness and direct
the reader to the appropriate sources. Often
these methods extend the accessible tem-
perature range for diffusion measurements.
In all cases specific atomicmodels need to
be introduced in order to extract a diffusion
coefficient. Inasmuch as there are inevita-
bly approximations in a model, the result-
ing diffusion coefficients may not be as re-
liable as those obtained from a concentra-
tion profile. In some cases additional corre-
lation information can be extracted, e.g.,
nuclear methods. The diffusion coefficients
found from these methods are essentially
“self” diffusion coefficients but chemical
diffusion coefficients for interstitial solutes
are obtained from the Gorsky effect.
3.5.3.1 Relaxation Methods
In the relaxation methods net atomic mi-
gration is due to external causes such as
stress of magnetic field (Nowick and Berry,
1972). The best known phenomena are (a)
the Gorsky effect (Gorsky, 1935); for a re-
view see Alefeld et al. (1970) and for an in-
troduction see Borg and Dienes (1988); (b)
the Snoek effect (Snoek, 1939); for an ex-www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

286. position see Wert (1970) and for an intro-
duction see Borg and Dienes (1988); and
(c) the Zener effect (Zener, 1947, 1951);
for an introduction see Bocquet et al.
(1996).
3.5.3.2 Nuclear Methods
A number of nuclear methods have be-
come of increasing importance for deter-
mining diffusion coefficients in solids. The
first of these is quasielastic neutron scat-
tering (QNS), which has often been used
for situations where the diffusion coeffi-
cients are larger than about 10
–11
m
2
s
–1
and the diffusing species exhibits a reason-
ably high scattering cross-section. Most
of the applications have been to hydrogen
(protonic) in metals (see, for example,
Janot et al., 1986). For a detailed introduc-
tion to the subject see Lechner (1983) and
Zabel (1984). The second method is nu-
clear magnetic resonance (NMR). NMR is
especially sensitive to interactions of the
nuclear moments with fields produced by
their local environments. Diffusion of a nu-
clear moment can cause variations in these
fields and can significantly affect the ob-
served resonance. In particular, diffusion
affects a number of relaxation times in
NMR. In favorable cases diffusion coeffi-
cients between 10
–18
and 10
–10
m
2
s
–1
are
accessible. For a detailed introduction to
the subject see Stokes (1984) and Heitjans
and Schirmer (1998). We make special
mention of pulsed field gradient (PFG)
NMR, which has been found to be espe-
cially useful for studying anomalous diffu-
sion (Kärger et al., 1998). Finally, we men-
tion Mössbauer spectroscopy (MBS), which
shows considerable promise for under-
standing diffusion processes in solids. The
general requirement is that the diffusion
coefficient of Mössbauer active isotope
should be larger than about 10
–13
m
2
s
–1
.
Most applications have been to solids in-
corporating the rather ideal isotope
57
Fe.
See Mullen (1984) for a detailed discus-
sion of MBS and Vogl and Feldwisch (1998)
for several recent examples of applica-
tions.
3.5.4 Surface Diffusion Methods
The techniques for measuring surface
diffusion are generally quite different from
solid-state techniques. For short distance,
microscopic or “intrinsic” diffusion (see
Sec. 3.2.2.5), the method of choice is the
field ion microscope. Much elegant work
has been carried out with this technique by
Erlich and co-workers; see for example Er-
lich and Stott (1980) and Erlich (1980).
With this method singleatoms can be im-
aged and followed. Another method, this
time using the field electron microscope
(Lifshin, 1992), correlates fluctuations of
an emission current from a very small area
with density fluctuations arising from
surface diffusion in and out of the probe
area (Chen and Gomer, 1979). Other meth-
ods include quasi-elastic scattering of
low energy He atoms (formally analogous
to quasi-elastic neutron scattering) and re-
laxation measurements, making use of dep-
osition of the adsorbate in a non-equilib-
rium configuration, followed by annealing
which permits relaxation to equilibrium.
Techniques useful for following the relaxa-
tion process at this microscopic level in-
clude pulsed molecular beam combined
with fast scanning IR interferometry
(Reutt-Robey et al., 1988) and work func-
tion measurements (Schrammen and Hölzl,
1983).
Many methods exist for long distance or
macrosopic diffusion. When the diffusing
species is deposited as a source and the ap-
propriate geometries are known then scan-
ning for the concentration profile followed
234 3 Diffusion Kinetics in Solidswww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

287. 3.7 References 235
by processing with the appropriate solution
to Fick’s Second Law (the Diffusion Equa-
tion) Eq. (3-5) will give the mass transfer
diffusion coefficient. Radioactive tracers
can certainly be used for this (Gjostein,
1970). The concentration profile (in the
case of hetero-diffusion) can also be deter-
mined using scanning SIMS, local XPS,
scanning Auger, scanning EM and scan-
ning STM (Bonzel, 1990). Of interest when
the atoms are weakly adsorbed is laser
induced thermal desorption (LITD) (Vis-
wanathan et al., 1982). The field electron
microscope can be used to image the diffu-
sion front of adatoms migrating into a re-
gion of clean surface (Gomer, 1958). Fi-
nally, the “capillary method”, probably the
best known of the macroscopic methods,
starts with a perturbed surface, usually in a
periodic way, say by sinusoidal grooving.
There is now a driving force to minimize
the surface Gibbs energy. The time depen-
dence of the decreasing amplitude of, say,
the sinusoidal profile, can be processed to
give the mass transfer diffusion coefficient;
see Bonzel (1990).
The reader is referred to a number of the
fine reviews in the area, e.g., Rhead
(1989), Bonzel (1990), and the book edited
by Vu Thien Binh (1983). The first of these
relates surface diffusion to a number of
technologically important processes such
as thin film growth, sintering, and cataly-
sis. The second is a comprehensive compi-
lation of surface diffusion data on metals
and a detailed survey of experimental
methods.
3.6 Acknowledgements
I thank my colleagues in diffusion from
whom I have learnt so much. This work
was supported by the Australian Research
Council.
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291. 4 Statistical Theories of Phase Transitions
Kurt Binder
Institut für Physik, Johannes Gutenberg-Universität Mainz,
Mainz, Federal Republic of Germany
List of Symbols and Abbreviations  . . . . . . . . . . . . . . . . . . . . . . . . . . 240
4.1 Introduction................................. 245
4.2 Phenomenological Concepts........................ 246
4.2.1 Order Parameters and the Landau Symmetry Classification . . . . . . . . . 246
4.2.2 Second-Order Transitions and Concepts about Critical Phenomena
(Critical Exponents, Scaling Laws, etc.) . . . . . . . . . . . . . . . . . . . 260
4.2.3 Second-Order Versus First-Order Transitions; Tricritical
and Other Multicritical Phenomena  . . . . . . . . . . . . . . . . . . . . . 269
4.2.4 Dynamics of Fluctuations at Phase Transitions  . . . . . . . . . . . . . . . 277
4.2.5 Effects of Surfaces and of Quenched Disorder on Phase Transitions:
a Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.3 Computational Methods Dealing with the Statistical Mechanics
of Phase Transitions and Phase Diagrams................ 283
4.3.1 Models for Order–Disorder Phenomena in Alloys  . . . . . . . . . . . . . 283
4.3.2 Molecular Field Theory and its Generalization
(Cluster Variation Method, etc.)  . . . . . . . . . . . . . . . . . . . . . . . 288
4.3.3 Computer Simulation Techniques  . . . . . . . . . . . . . . . . . . . . . . 293
4.4 Concepts About Metastability....................... 298
4.5 Discussion.................................. 303
4.6 References.................................. 304
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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292. List of Symbols and Abbreviations
A prefactor of Friedel interaction
a lattice spacing
a
i nearest-neighbor distance
Aˆ, Aˆ¢ critical amplitude of specific heats 
B critical amplitude of the order parameter
c concentration
C specific heat
C phenomenological coefficient
c lattice spacing in z-direction
c cluster of geometric configuration
Dc concentration difference
Cˆ, Cˆ¢ critical amplitudes of ordering susceptibilities
c
coex concentration at coexistence curve
c
i concentration of lattice site i
c
ijk… elastic constants
d dimensionality
Dˆ critical amplitude at the critical isotherm
d* marginal dimensionality
E electric field
e
i random vectors
e
l(k,x) phonon polarization vector
F[
F(x)] Helmholtz energy function in operator form
f factor of order unity
F Helmholtz energy
DF* Helmholtz energy barrier
F(
j) Helmholtz energy functional
F
˜
scaling function of the Helmholtz energy
f, f
I
,…,f
IV
,fcoefficients
F
0 background Helmholtz energy of disordered phase
f
cg coarse-grained Helmholtz energy
f
mn quadrupole moment tensor
F
reg background term of the Helmholtz energy
g constant factor in the structure function
G reciprocal lattice vector
G(x) order parameter correlation function
G
˜
(x/
x) scaling function of the correlation function
Gˆ critical amplitude of the correlation function
G(x,
x) order parameter correlation function
g
nc multispin correlation function
H ordering field
H magnetic field
∫ Hamiltonian
H
˜
scaled ordering field
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293. List of Symbols and Abbreviations 241
h, h
c uniform magnetic field, critical value
H
R random field (due to impurities)
i number and index for different lattice sites
J energy gained when two like atoms occupy neighboring sites
J exchange interaction
J
˜
Fourier transform of the exchange interaction
J
ij exchange interaction between spins at sites iand j(in cases where it is ran-
dom)
J
m strength of the magnetic interaction
J
nn nearest-neighbor interaction
J
nnn next-nearest-neighbor interaction
k wavevector of phonon
K
1, K
2 phenomenological coefficients
k
B Boltzmann constant
k
F Fermi wavenumber
L coarse-graining length
Lagrangian
M period (in Fig. 4-10)
M magnetization
M
l mass of an atom
m
m sublattice magnetization
M
s spontaneous magnetization
N degree of polymerization
n
a lattice sites in state a
p pressure
P polarization
p dielectric polarization
P(J) statistical distribution of random variable J
P({S
i}) probability that a configuration {S
i} occurs
p
+ probability for spin up
p
– probability for spin down
p
b parameter in Landau equation (bicritical points)
Q normal coordinate
q
a primitive vectors of the reciprocal lattice
q
max value of q where  c(q) is at a maximum
q* nonzero wavenumber
q
EA ordering parameter for the Ising spin glass
r phenomenological coefficient in the Landau expansion
R effective range of interaction
R ratio of J
nnand J
nnn
R* critical radius of a droplet
R
l
i
site of atom lin unit cell i
r, r¢ coefficient in the Landau equation
r
c radius of polymer coil
S entropywww.iran-mavad.com
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294. S(q) structure factor
S(q,
w) scattering function
S
i, S
j unit vectors in direction of the magnetic moment
t time
t reduced temperature 1–T/T
0(in Sec. 4.2.2)
T
0 temperature where rchanges sign
T
1 temperature where metastable ordered phase vanishes
T
b bicritical point
T
c critical temperature
t
cr reduced temperature at crossover
T
L Lifshitz point
T
l(x) transition temperature of superfluid 
4
He
T
t temperature of tricritical point
u phenomenological coefficient
u coefficient in Landau equation
U enthalpy
u
l displacement vector
U
d surface of a d-dimensional sphere
v coefficient in the Landau equation
v(|x|) Friedel potential
V total volume of the thermodynamic system
V(x
i–x
j) pairwise interaction of two particles
V
ab interaction between atoms A and B
w coefficient in the Landau equation
W action in classical mechanics
x, x
n, x
m, x
aposition vector and its components
z dynamical critical exponent
z coordinate (of position vector)
z coordination number
Z partition function
scaling variable
a
1m, a
2 notation for phases in phase diagram
a
m magnetic phase
a
n nonmagnetic phase
a(x) short-range order parameter for metal alloys
a,a¢ critical exponents related to specific heat
b exponent related to the order parameter
G
ˆ
critical amplitude of the susceptibility
g,g¢ critical exponents related to the susceptibility
G
0 phenomenological rate factor
d critical exponent at the critical isotherm
d
mn Kronecker symbol
e binding energy of a particle
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295. List of Symbols and Abbreviations 243
e
ik strain tensor
h exponent related to the decay of correlations at T
c
l wavelength
l label of the phonon branch
L factor changing the length scale
m chemical potential
m magnetic moment per spin
D
m chemical potential difference
m
coex chemical potential at phase coexistence
n exponent related to the correlation length
x correlation length
x
i concentration difference (Eq. (4-126))
r density
r(x) charge density distribution function
t characteristic time
j phase shift in Friedel potential
F order parameter
F(x) order parameter density
F
i concentration difference between two sublattices
F
ms order parameter of metastable state
F
s spinodal curve
c(q) wavevector-dependent susceptibility
c
el dielectric tensor
c
T response function of the order parameter
c
T isothermal susceptibility
c
T staggered susceptibility
y concentration difference between two sublattices
y
I,II
ordering parameter for D0
3structure
y
a amplitude of mass density waves
w characteristic frequency (for fluctuations)
w˜ scaling function
AF antiferromagnetic
cg coarse-grained (as index)
coex coexistent (as index)
CV cluster variation method
DAG dysprosium aluminum garnet
ESR electron spin resonance
f.c.c. face-centered cubic
LRO long-range order
MC Monte Carlo (method)
MD molecular dynamics
MFA mean field approximation
MFA molecular field approximationwww.iran-mavad.com
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296. ms metastable state (as index)
NMR nuclear magnetic resonance
P paramagnetic
sc simple cubic
SRO short-range order
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297. 4.1 Introduction 245
4.1 Introduction
This chapter deals with the general theo-
retical concepts of phase transitions of ma-
terials and their basis in statistical thermo-
dynamics. This field has seen extensive
scientific research over many decades, and
a vast amount of literature exists, e.g., the
statistical theory of critical point phenom-
ena fills a series of monographs that so far
consists of 17 volumes (Domb and Green,
1972–1976; Domb and Lebowitz, 1983–
1997). Introductory texts require whole
books (Stanley, 1971; Ma, 1976; Patashin-
skii and Pokrovskii, 1979; Yeomans, 1992),
and so does a comprehensive account of
the Landau theory of phase transitions 
(Tolédano and Tolédano, 1987).
This chapter therefore cannot give an ex-
haustive description of the subject; instead
what is intended is a tutorial overview,
which gives the flavor of the main ideas,
methods, and results, with emphasis on the
aspects which are particularly relevant for
materials science, and hence may provide a
useful background for other chapters in this
book. Hence this chapter will contain
hardly anything new for the specialist; for
the non-specialist it will give an initial or-
ientation and a guide for further reading.
To create a coherent and understandable
text, a necessarily arbitrary selection of
material has been made which reflects the
author’s interests and knowledge. Reviews
containing complementary material have
been written by De Fontaine (1979), who
emphasizes the configurational thermody-
namics of solid solutions, and by Khacha-
turyan (1983), who emphasizes the theory
of structural transformations in solids 
(which, from a different perspective, are
discussed by Bruce and Cowley, 1981).
There are also many texts which either con-
centrate on phase transformations of par-
ticular types of materials, such as alloys
(Tsakalakos, 1984; Gonis and Stocks, 1989;
Turchi and Gonis, 1994), magnetic materi-
als (De Jongh and Miedema, 1974; Ausloos
and Elliott, 1983), ferroelectrics (Blinc and
Zeks, 1974; Jona and Shirane, 1962), liquid
crystals (Pershan, 1988; De Gennes, 1974;
Chandrasekhar, 1992), polymeric materials
(Flory, 1953; De Gennes, 1979; Riste and
Sherrington, 1989; Binder, 1994; Baus et
al., 1995), etc., or on particular types of
phase transitions, such as commensurate–
incommensurate transitions (Blinc and Le-
vanyuk, 1986; Fujimoto, 1997), multicriti-
cal transitions (Pynn and Skjeltorp, 1983),
first-order transitions (Binder, 1987a),
martensitic transformations (Nishiyama,
1979; Salje, 1990), glass transitions 
(Jäckle, 1986; Angell and Goldstein, 1986;
Cusack, 1987; Zallen, 1983; Zarzycki,
1991), percolation transitions (Stauffer and
Aharony, 1992), melting (Baus, 1987),
wetting transitions (Dietrich, 1988; Sulli-
van and Telo da Gama, 1985), metal–insu-
lator transitions (Mott, 1974; Friedmann
and Tunstall, 1978), etc. Other approaches
focus on particular ways of studying phase
transitions: from the random-phase approx-
imation (Brout, 1965) and the effective-
field theory (Smart, 1966) to advanced
techniques such as field-theoretical (Amit,
1984) or real space renormalization (Burk-
hardt and van Leeuwen, 1982), and com-
puter simulation studies of phase transi-
tions applying Monte Carlo methods 
(Binder, 1979, 1984a; Mouritsen 1984;
Binder and Heermann, 1988; Binder and
Ciccotti, 1996) or the molecular dynamics
techniques (Ciccotti et al., 1987; Hockney
and Eastwood, 1988; Hoover, 1987; Binder
and Ciccotti, 1996), etc. For particular
models of statistical mechanics, exact de-
scriptions of their phase transitions are
available (Baxter, 1982). Of particular
interest also are the dynamics of critical
phenomena (Enz, 1979) and the dynamicswww.iran-mavad.com
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298. of first-order phase transitions (Gunton 
et al., 1983; Koch, 1984; Binder, 1984b,
1989; Haasen et al., 1984; Zettlemoyer,
1969; Abraham, 1974; Gunton and Droz,
1983; Herrmann et al., 1992). The latter
subject is treated in two other chapters 
in this book (see the Chapters by Wagner 
et al. (2001) and Binder and Fratzl (2001))
and will not be considered here.
In this chapter, we shall briefly discuss
the statistical thermodynamics of phase
transitions on a phenomenological macro-
scopic level (Sec. 4.2), i.e., the Landau the-
ory of first- and second-order transitions,
critical and multicritical phenomena, and
also the dynamics of fluctuations at phase
transitions, and the effects of quenched 
disorder in solids. The second part (Sec.
4.3) is devoted to the complementary “mi-
croscopic” approach, where we start from
“model Hamiltonians” for the system under
consideration. The statistical mechanics of
such models naturally leads to the consider-
ation of some computational methods, i.e.,
methods based on the molecular field the-
ory and its generalizations, and methods
based on computer simulation techniques.
Since in solid materials metastable phases
are ubiquitous, e.g., diamond is the meta-
stable modification of solid carbon whereas
graphite is the stable phase, some com-
ments about the statistical mechanics of
metastability are made in Sec. 4.4.
4.2 Phenomenological Concepts
In this section, the main facts of the the-
ory of phase transitions are summarized,
and the appropriate terminology intro-
duced. Rather than aiming at completeness,
examples which illustrate the spirit of the
main approaches will be discussed, includ-
ing a discussion of critical phenomena and
scaling laws.
4.2.1 Order Parameters and the Landau
Symmetry Classification
Table 4-1 lists condensed matter systems
that can exist in several phases, depending
on external thermodynamics parameters
such as pressure, p, temperature, T, electric
or magnetic fields (E,H). We assume that
an extensive thermodynamic variable can
be identified (i.e., one that is proportional
246 4 Statistical Theories of Phase Transitions
Table 4-1.Order parameters for phase transitions in various systems.
System Transition Order parameter
Liquid–gas Condensation/evaporation Density difference D
r=r
l–r
g
Binary liquid mixture Unmixing Composition difference Dc=c
(2)
coex
–c
(1)
coex
Nematic liquid Orientational ordering –
1
2
·3(cos
2
q)–1Ò
Quantum liquid Normal fluid  asuprafluid ·
yÒ(y: wavefunction)
Liquid–solid Melting/crystallization r
G(G= reciprocal lattice vector)
Magnetic solid Ferromagnetic ( T
c) Spontaneous magnetization M
Antiferromagnetic (T
N) Sublattice magnetization M
s
Solid binary mixture Unmixing Dc=c
(2)
coex
–c
(1)
coex
AB Sublattice ordering y=(Dc
II
–Dc
I
)/2
Dielectric solid Ferroelectric ( T
c) Polarization P
Antiferroelectric (T
N) Sublattice polarization P
s
Molecular crystal Orientational ordering Y
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299. 4.2 Phenomenological Concepts 247
to the volume of the system) which distin-
guishes between these phases (examples
are also given in Table 4-1), called the “or-
der parameter”. We shall denote the order
parameter as 
F, and the conjugate thermo-
dynamic variable, the “ordering field”, as
H. Using the thermodynamic potential, F,
which has as “natural variables” a field, H
(which is an “intensive” thermodynamic
variable, i.e., independent of the volume),
and the temperature, T, we have
F= – (∂F/∂H)
T (4-1)
the other derivative being the entropy,
S= – (∂F/∂T)
H (4-2)
As examples, consider a ferromagnet, where
the order parameter is the magnetization,
M, or a ferroelectric, where the order pa-
rameter is the dielectric polarization, P,
M= – (∂F/∂H)
T (4-3a)
P = – (∂F/∂E)
T (4-3b)
It is clear that such thermodynamic rela-
tions can be written for any material, but a
quantity qualifies as an “order parameter”
when a particular value of the “ordering
field” exists where the order parameter ex-
hibits a jump singularity between two dis-
tinct values (Fig. 4-1). This means that for
these values of the ordering field a first-or-
der phase transition occurs, where a first
derivative of the thermodynamic potential
Fexhibits a singularity. At this transition,
two phases can coexist; e.g., at the liq-
uid–gas transition for a chemical potential
m=m
coex(T), two phases with different
density coexist; and in a ferromagnet at
zero magnetic field, phases with opposite
sign of spontaneous magnetization can co-
exist. Although the fluid-magnet analogy
(Fig. 4-1) goes further, since the first-order
lines in the (
m, T) or (H , T) plane in both
cases end in critical points which may even
be characterized by the same critical expo-
nents (see below), there is also an impor-
tant distinction that in the magnetic prob-
lem, the Hamiltonian possesses a symme-
try with respect to the change of sign of the
magnetic field; reversing this sign and also
reversing the sign of the magnetization
leave the Hamiltonian invariant. Because
of this symmetry, the transition line must
occur at H = 0. Conversely, if the system 
Figure 4-1.The fluid-magnet analogy. On varying
the chemical potential 
mat m
(T)
coex
, the density rjumps
from the value at the gas branch of the gas–liquid co-
existence curve (
r
gas=r
(1)
coex
) to the value at the liquid
branch of the coexistence curve (
r
liquid=r
(2)
coex
) (top
left). Similarly, on varying the (internal) magnetic
field H, the magnetization Mjumps from the nega-
tive value of the spontaneous magnetization (–M
s) to
its positive value (top right). While this first-order
liquid–gas transition occurs at a curve 
m
coex(T) in
the 
m–T-plane ending in a critical point (m
c,T
c)
where the transition is then of second order, the curve
where phases with positive and negative spontaneous
magnetization can coexist simply is H=0 (T<T
c)
(middle). The order parameter (density difference
D
r, or spontaneous magnetization M
s) vanishes ac-
cording to a power law near T
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300. at H= 0 is in a monodomain state with 
either positive or negative spontaneous
magnetization, this symmetry is violated;
this is described as “spontaneous symmetry
breaking”. No such obvious symmetry, on
the other hand, is identified for the liq-
uid–gas transition of fluids. Consequently,
the curve 
m
coex(T) is a nontrivial function
in the 
m–T-plane, and no simple symmetry
operation acting on the gas phase atoms ex-
ists that would transform this phase into a
liquid, or vice versa. Similar lack of sym-
metry between the phases is noted for un-
mixing transitions in binary fluid or solid
mixtures, where the order parameter is a
concentration difference, Dc =c
(2)
coex
–c
(1)
coex
,
cf. Table 4-1, whereas solid binary mix-
tures which exhibit sublattice ordering do
possess a symmetry. The order parameter in
an alloy such as brass (b-CuZn) is the dif-
ference between the relative concentration
of the two sublattices, 
y=(Dc
II
–Dc
I
)/2.
However, the two sublattices physically are
completely equivalent; therefore, the Ha-
miltonian possesses a symmetry against the
interchange of the two sublattices, which
implies that 
ychanges sign, just as the
(idealized!) ferromagnet does for H=0
(Fig. 4-1). In this example of an alloy
which undergoes an order–disorder transi-
tion where the permutation symmetry
between the two sublattices is spontane-
ously broken, the “ordering field” conju-
gate to the order parameter is a chemical
potential difference between the two sub-
lattices, and hence this ordering field is 
not directly obtainable in the laboratory.
The situation is comparable to the case 
of simple antiferromagnets, the order pa-
rameter being the “staggered magnetiza-
tion” (= magnetization difference between
the sublattices), and the conjugate ordering
field would change sign from one sublat-
tice to the other (“staggered field”). Al-
though the action of such fields usually
cannot be measured directly, they never-
theless provide a useful conceptual frame-
work.
Another problem which obscures the
analogy between different phase transitions
is the fact that we do not always wish to
work with the corresponding statistical en-
sembles. For a liquid–gas transition, we
can control the chemical potential via the
fluid pressure, and thus a grand-canonical
ensemble description makes sense. How-
ever, for the binary mixture (AB), the or-
dering field would be a chemical potential
difference D
m=m
A–m
Bbetween the spe-
cies. In an ensemble where D
mis the exter-
nally controlled variable A atoms could
transform into B, and vice versa. Experi-
mentally, of course, we do not usually have
mixtures in contact with a reservoir with
which they can exchange particles freely,
instead mixtures are kept at a fixed relative
concentration c. (Such an equilibrium with
a gas reservoir can only be realized for
interstitial alloys such as metal hydrides or
oxides.) Thus the experiment is described
by a canonical ensemble description, with
(T,c) being the independent variables, al-
though the grand-canonical ensemble of
mixtures is still useful for analytical theo-
ries and computer simulation. Now, as is
obvious from Fig. 4-1, in a canonical en-
semble description of a fluid, the first-
order transition shows up as a two-phase
coexistence region. Here, at a given den-
sity, 
r, with r
gas<r<r
liquid, the relative
amounts of the two coexisting phases are
given by the lever rule if interfacial contri-
butions to the thermodynamic potential can
be neglected. The same is true for a binary
mixture with concentration cinside the two-
phase coexistence region, c
(1)
coex
<c<c
(2)
coex
.
These different statistical ensembles also
have pronounced consequences on the dy-
namic properties of the considered sys-
tems: in binary mixtures at constant con-
248 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
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301. 4.2 Phenomenological Concepts 249
centration, and also in fluids at constant
density, the order parameter is a conserved
quantity, whereas for a fluid at constant
chemical potential, the order parameter is
not conserved. The latter situation occurs,
for instance, for fluid–gas transitions in
physisorbed monolayers at surfaces, which
can exchange molecules with the surround-
ing gas phase which is in equilibrium with
this adsorbed layer.
An important distinction to which we
turn next is the order of a phase transition.
In the examples shown in the upper part of
Fig. 4-1, a first derivative of the appropri-
ate thermodynamic potential has a jump
singularity and therefore such transitions
are called first-order transitions.However,
if we cool a ferromagnet down from the
paramagnetic phase in zero magnetic field,
the spontaneous magnetization sets in con-
tinuously at the critical temperature T
c
(lower part of Fig. 4-1). Similarly, on cool-
ing the alloy b-CuZn from the state where
it is a disordered solid solution (in the
body-centered cubic phase), sublattice or-
dering sets in continuously at the critical
temperature (T
c= 741 K, see Als-Nielsen
(1976)). Whereas the first derivatives of
the thermodynamic potential at these con-
tinuous phase transitions are smooth, the
second derivatives are singular, and there-
fore these transitions are also called sec-
ond-order transitions. For example, in a
ferromagnet the isothermal susceptibility
c
Tand the specific heat typically have
power law singularities (Fig. 4-2)
(4-4)
(4-5)
CTFT
ATT T T
ATT TT
HH
TT≡− ∂ ∂
−>
′−<
⎧
⎨
⎩
→
−
−′
(/)
ˆ
(/ ) ,
ˆ
(/),
22
0
1
1
=
cc
cc
=
c
a
a
c
g
g
TT
TT FH
CTT T T
CTT TT
≡− ∂ ∂
−>
′−<
⎧
⎨
⎩
→
−
−′
(/ )
ˆ
(/ ) ,
ˆ
(/),
22
1
1
=
c
cc
cc
In this context (see also Fig. 4-1), a, a¢, b,
g, and g¢are critical exponents, while Aˆ, Aˆ¢,
Bˆ, Cˆ, and C ˆ¢are called critical amplitudes.
Note that Bˆand 
brefer to the spontaneous
magnetization, the order parameter (Fig. 4-1)
(4-6)
Behavior of the specific heat such as de-
scribed by Eq. (4-5) immediately carries
over to systems other than ferromagnets
such as antiferromagnets, the liquid–gas
system near its critical point, and brass
near its order–disorder transition; we must
remember, however, that Hthen means the
MBTT
TT
sc
=
c→
−
ˆ
(/)1
b
Figure 4-2.Schematic variation with temperature T
plotted for several quantities near a critical point T
c:
specific heat C
H(top), ordering “susceptibility” c
T
(middle part), and correlation length xof order pa-
rameter fluctuations (bottom). The power laws which
hold asymptotically in the close vicinity of T
care in-
dicated.www.iran-mavad.com
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302. appropriate ordering field. In fact, this is
also true for Eq. (4-4), but then the physical
significance of 
c
Tchanges. For a two-sub-
lattice antiferromagnet, the ordering field
is a “staggered field”, which changes sign
between the two sublattices, and hence is
thermodynamically conjugate to the order
parameter of the antiferromagnet. Al-
though such a field normally cannot be ap-
plied in the laboratory, the second deriva-
tive, 
c
T(in this case it is called “staggered
susceptibility”) is experimentally access-
ible via diffuse magnetic neutron scatter-
ing, as will be discussed below. Similarly,
for the ordering alloy b-CuZn, Hstands for
a chemical potential difference between the
two sublattices; the response function 
c
Tis
again physically meaningful and measures
the peak intensity of the diffuse scattering
of X-rays or neutrons. (This scattering
peak occurs at the superlattice Bragg spot
characteristic of the sublattice ordering
considered.) Finally, in the gas–liquid
transition considered in Fig. 4-1, His the
chemical potential and 
c
Tthe isothermal
compressibility of the system.
As will be discussed in more detail in
Sec. 4.2.2, the divergences of second deriv-
atives of the thermodynamic potential at a
critical point (Eqs. (4-4), (4-5), Fig. 4-2)
are linked to a diverging correlation length
of order-parameter fluctuations (Fig. 4-2).
Hence any discussion of phase transitions
must start with a discussion of the order pa-
rameter. The Landau theory (Tolédano and
Tolédano, 1987) which attempts to expand
the thermodynamic potential in powers of
the order parameter, gives a first clue to the
question of whether a transition is of sec-
ond or first order (see Sec. 4.2.3).
We first identify the possible types of or-
der parameters, since this will distinctly af-
fect the nature of the Landau expansion. In
Eqs. (4-1) and (4-4) to (4-6) we have
treated the ordering field Hand the order
parameter 
Fas scalar quantities; although
this is correct for the gas–fluid transition
and for the unmixing transition of binary
mixtures, the uniaxial ferro- or antiferro-
magnets, uniaxial ferro- or antiferroelec-
trics, etc., and for order–disorder transi-
tions in alloys or mixed crystals (solid so-
lutions), when only two sublatties need to
be considered, there are also cases where
the order parameter must have a vector or
tensor character. Obviously, for an iso-
tropic ferromagnet or isotropic ferroelec-
tric the order parameter in the three-dimen-
sional space is a three-component vector
(see Eq. (4-3)). It also makes sense to con-
sider systems where a planar anisotropy is
present, such that Mor E(Eq. (4-3)) must
lie in a plane and hence a two-component
vector applies as an order parameter. How-
ever, for describing antiferromagnetic or-
dering and for order–disorder transitions
with many sublattices multicomponent or-
der parameters are also needed, and the
number of components of the order param-
eter, the so-called “order parameter dimen-
sionality”, is dictated by the complexity of
the structure, and has nothing to do with
the spatial dimension. This is best under-
stood by considering specific examples.
Consider, for example, the ordering of
Fe–Al alloys (Fig. 4-3): whereas in the dis-
ordered A2 phase Fe and Al atoms are ran-
domly distributed over the available lattice
sites, consistent with the considered con-
centration (although there may be some
short-range order), in the ordered B2 phase
(the FeAl structure) the bcc lattice is split
into two inter-penetrating simple cubic (sc)
sublattices, one taken preferentially by Fe,
the other by Al atoms. This is the same
(one-component) ordering as in b-CuZn.
What is of interest here is the further “sym-
metry breaking” which occurs for the D0
3
structure (realized in the Fe
3Al phase):
here four face-centered cubic (f.c.c.) sub-
250 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
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303. 4.2 Phenomenological Concepts 251
lattices a, b, c, d must be distinguished. Al-
though the concentrations on sublattices a
and c are still equivalent, a further symme-
try breaking occurs between the sublattices
b and d, such that only one sublattice (e.g.,
d) is preferentially occupied by Al, all
other sublattices being preferentially occu-
pied by iron. However, the role of the sub-
lattices (a, c) and (b, d) can be inter-
changed; hence a two-component order pa-
rameter is needed to describe the structure,
namely (see for example, Dünweg and
Binder (1987))
(4-7)
where c
iis the concentration of lattice site i
and m
mthe sublattice “magnetization” in
pseudo-spin language. As can example of
an ordering with m= 8 components, we can
consider the f.c.c. antiferromagnet MnO:
as Fig. 4-4 shows, the magnetic structure
consists of an antiferromagnet arrangement
of ferromagnetically ordered planes. If the
ordering were uniaxially anisotropic, we
y
y
m
m
I
acbd
II
acbd
=
=
mmmm
mmmm
mN c
i
i
−+−
−++−
≡−
∑(/ ) ( )121
Œ
Figure 4-3.Body-centered cubic lattice showing the
B2 structure (top) and D0
3structure (bottom). The
top part shows assignments of four sublattices, a, b,
c, and d. In the A2 structure, the average concentra-
tions of A and B atoms are identical on all four sub-
lattices whereas in the B2 structure the concentra-
tions at the b and d sublattices are the same (example:
stoichiometric FeAl, with Fe on sublattices a and c;
Al on sublattices b and d). In the D0
3structure, the
concentrations at the a and c sublattice are still the
same, whereas the concentration at sublattice b dif-
fers from the concentration at sublattice d (example:
stoichiometric Fe
3Al, with Al on sublattice b; all
other sublattices taken by Fe). From Dünweg and
Binder (1987).
B2
D0
3
Figure 4-4.Schematic view of the MnO structure,
showing the decomposition of the fcc lattice into two ferromagnetic sublattices with opposite orientation of the magnetization (indicated by full and open cir- cles, respectively). Each sublattice is a stack of par- allel close-packed planes (atoms form a triangular lattice in each plane). Note that there are four equiv- alent ways in which these planes can be oriented in an fcc crystal.www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

304. would have m= 4 because of the four pos-
sible ways of orienting the planes (and
there could be eight kinds of different do-
mains coexisting in the system). Since for
MnO the order parameter can take any or-
ientation is a plane, it has two independent
components, and hence the total number of
order parameter components is m= 8 (Mu-
kamel and Krinsky, 1976). An even larger
number of components is needed to de-
scribe the ordering of solid 
3
He. In view of
this, it is clear that even mÆ•, which is
also called the spherical model (Berlin and
Kac, 1952; Joyce, 1972) is a useful limit to
consider from the theoretical point of view.
Apart from this m -vector model for the
order parameter, Table 4-1 illustrates the
need to consider order parameters of tenso-
rial character. This happens in molecular
crystals such as para-H
2(Fig. 4-5), N
2, O
2
and KCN, in addition to liquid crystals.
Whereas the atomic degrees of freedom
considered for ferro- or antiferromagnetic
order are magnetic dipole moments, for
ferro- or antiferroelectric order electric di-
pole moments, the degree of freedom of the
molecules in Fig. 4-5 which now matters is
their electric quadrupole moment tensor:
(4-8)
where 
r(x) is the charge density distribu-
tion function of a molecule, x=(x
1, x
2, x
3),
and 
d
mnis the Kronecker symbol.
Proper identification of the order param-
eter of a particular system therefore needs a
detailed physical insight, and often is com-
plicated because of coupling between dif-
ferent degrees of freedom, e.g., a ferromag-
netic material which is cubic in the para-
magnetic phase may become tetragonal as
the spontaneous magnetization develops,
owing to magnetostrictive couplings (for a
more detailed discussion of this situation,
fx xx
mn m n
l
lmn rd=d
=
∫ ∑ −
⎛
⎝
⎜
⎞
⎠
⎟
xx()
1
3
1
3
2
see Grazhdankina (1969)). In this example
it is clear that the spontaneous magnetiza-
tion, M, is the “primary order parameter”
whereas the tetragonal distortion (c/a–1),
where cis the lattice spacing in the lattice
direction where the magnetization direc-
tion occurs and ais the lattice spacing 
perpendicular to it, is a “secondary order
parameter”. For purely structural phase
transitions where all considered degrees 
of freedom are atomic displacements, the
proper distinction between primary and
secondary order parameters is much more
subtle (Tolédano and Tolédano, 1987).
We first formulate Landau’s theory for
the simplest case, a scalar order parameter
density 
F(x). This density is assumed to
be small near the phase transition and
slowly varying in space. It can be obtained
by averaging a microscopic variable over a
suitable coarse-graining volume L
d
in d-di-
mensional space. For example, in an aniso-
tropic ferromagnet the microscopic vari-
able is the spin variable 
F
i= ±1 pointing in
the direction of the magnetic moment at
lattice site i; in an ordering alloy such as b-
CuZn the microscopic variable is the dif-
ference in concentration of Cu between the
two sublattices I and II in unit cell i, 
F
i=
c
i
II–c
i
I; and in an unmixing alloy such as
Al–Zn, the microscopic variable is the
concentration difference between the local
Al concentration at lattice site i(c
i=1 if it
252 4 Statistical Theories of Phase Transitions
Figure 4-5.Schematic view of a lattice plane of a
para-H
2crystal, indicating the orientational order of
the ellipsoid H
2molecules.www.iran-mavad.com 
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305. 4.2 Phenomenological Concepts 253
is taken by Al and c
i= 0 otherwise) and the
critical concentration, 
F
i=c
i–c
crit. In all
of these cases 
F(x) is considered to be an
order parameter field defined in continuum
space:
(4-9)
xbeing the center of gravity of the “vol-
ume” L
d
. The appropriate magnitude of the
linear dimension Lof the coarse-graining
cell will be discussed later; obviously it
must be much larger than the lattice spac-
ing in order for the continuum description
to make sense. Then a Helmholtz energy
functional F[
F(x)] is assumed:
where F
0is the background Helmholtz en-
ergy of the disordered phase, and r, u, and
Rare phenomenological constants. (In fact,
Rcan be interpreted as the effective range
of interaction between the atomic degrees
of freedom 
F
i, as will be seen below.) Ob-
viously, Eq. (4-10) is the Taylor series-type
expansion of F[
F(x)] in powers of F(x)
and —
F(x), where just the lowest order
terms were kept. This makes sense if both
the coefficients uand R
2
are positive con-
stants at T
c, whereas the essential assump-
tionwhich defines
Fas playing the role of
an order parameter of a second-order phase
transition is that the coefficient rchanges
its sign at the transition as the variable of
interest (the temperature in the present
case) is varied,
k
BTr= r¢(T– T
c) (4-11)
Note also that in Eq. (4-10) we have as-
sumeda symmetry in the problem against
1
1
2
1
4
1
2
0
24
2
kT
F
F
kT
ru
H
kT d
R
BB
B
= (4-10)
d
[()]
() ()
() [ ()]F
FF
FFx
xx x
xx
++
⎧
⎨
⎩
−+∇
⎫
⎬
⎭
∫
FF() /x=
iL
i
d
d
L
Œ
∑
the change of sign of the order parameter
for H= 0 and thus odd powers such as
F
3
(x) do not occur; this is true for magnets
(no direction of the magnetization is pre-
ferred without magnetic field in a ferro-
magnet) and for sublattice ordering of al-
loys such as b-CuZn (since whether the Cu
atoms preferentially occupy sublattices a, c
in Fig. 4-3 or sublattices b, d is equivalent),
but it is not true in general (e.g., third-order
terms do occur in the description of the
Cu
3Au structure, or in the ordering of rare
gas monolayers adsorbed on graphite in 
the ÷
–
3 structure, as will be discussed be-
low).
In order to understand the meaning and
use of Eq. (4-10), consider first the fully
homogeneous case, —
F(x)∫0, F(x)∫F
0;
then F[
F] is the standard Helmholtz en-
ergy function of thermodynamics, which
needs to be minimized with respect to 
Fin
order to determine the thermal equilibrium
state, and Údx=Vthe total volume of the
system. Thus,
(4-12)
which is solved by
F
0= 0, T> T
c
F
0= ± (–r/u)
1/2
(4-13)
= ± (r¢/k
Bu)
1/2
(T
c/T–1)
1/2
,T< T
c
Hence Eqs. (4-10) and (4-12) indeed yield
a second-order transition as Tis lowered
through T
c. For T<T
c, a first-order transi-
tion as a function of Hoccurs, since 
F
0
jumps from (–r/u)
1/2
to – (–r/u)
1/2
as H
changes sign. This behavior is exactly that
shown schematically in Fig. 4-1, with
b=1/2, Bˆ=(r¢/k
Bu)
1/2
and F
0(H=0)=M
S.
If u< 0 in Eq. (4-10), however, we must
not stop the expansion at fourth order but
rather must include a term –
1
6
vF
6
(x) (as-
suming now v> 0). Whereas in the second-
1
0
0
0
00
3kTV
F
ru
T
H
B
=
==
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟ +
F
FFwww.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

306. order case F ( F) has two minima for T<T
c
which continuously merge as TÆT
cand
only one minimum at 
F= 0 remains for
T>T
c(Fig. 4-6a) F [ F] now has three min-
ima for T
0<T<T
c, and the temperature T
0
where rchanges sign (r=r¢(T–T
0) now)
differs from the phase transition tempera-
ture T
cwhere the order parameter jumps
discontinuously from zero for T>T
cto
(
F
0)
T
c
=±(3u/4v)
1/2
, see Fig. 4-6b.
These results are found by analogy with
Eq. (4-12) from
(4-14)
which is solved by
(choosing the minus sign of the square root
would yield the maxima rather than the
minima in Fig. 4-6b). On the other hand,
we know that [F–F(0)]/(Vk
BT) = 0 in the
disordered phase, and T
ccan be found (Fig.
4-6b) from the requirement that the free
energy of the ordered phase then is equal to
F
0
22 22=−+ −uur/( ) ( / ) /vvv
1
0
0
0
00
2
0
4kTV
F
ru
T
H
B
=
=+=
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
+
F
FFF
() v
this value, i.e.,
(4-15)
With some simple algebra Eqs. (4-14) and
(4-15) yield
T
c= T
0+ 3u
2
/(32r¢v) (4-16)
and the “stability limit”, where the mini-
mum describing the metastable ordered
phase in the disordered phase above T
cdis-
appears, is given by
T
1= T
0+ u
2
/(8r¢v) (4-17)
The significance of such metastable states
as described by the Landau theory for first-
order transitions and the associated stabil-
ity limits T
0(for the disordered phase at
T<T
c) and T
1will be discussed further in
Sec. 4.4.
The alternative mechanism by which a
first-order transition arises in the Landau
theory with a scalar order parameter is 
the lack of symmetry of Fagainst a sign
change of 
F. Then we may add a term
–
1
3
wF
3
to Eq. (4-10), with another pheno-
[ ( ) ( )]/( )FFVkT
ru
TT
F
FFF
0
0
2
0
2
0
4 0
24 6
0
−
+
⎛
⎝
⎞
⎠
B
=
=+=
c
v
254 4 Statistical Theories of Phase Transitions
Figure 4-6.Schematic variation of the Helmholtz energy in the Landau model at transitions of (a) second or-
der and (b) first order as a function of the (scalar) order parameter 
F. Cases (a) and (b) assume a symmetry
around 
F= 0, whereas case (c) allows a cubic term.www.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

307. 4.2 Phenomenological Concepts 255
menological coefficient, w. For u> 0, F( F)
may have two minima (Fig. 4-6c); again
the transition occurs when the minima are
equally deep. For r=r¢(T–T
0) this hap-
pens when
T
c= T
0+ 8w
2
/(81ur¢) (4-18)
the order parameter jumping there from
F
0=–9r/wto  F
0= 0. Again a stability
limit of ordered state in the disordered
phase occurs, i.e.
T
1= T
0+ w
2
/(4ur¢) (4-19)
At this point, an important caveat should be
emphasized: free energy curves involving
several minima and maxima as drawn in
Fig. 4-6 are used so often in the literature
that many researchers believe these con-
cepts to be essentially rigorous. However,
general principles of thermodynamics indi-
cate that in thermal equilibrium the ther-
modynamic potentials are convex func-
tions of their variables. In fact, F (
F
0)
should thus be convex as a function of 
F
0,
which excludes multiple minima! For Fig.
4-6a and b this means that for T<T
cin
states with –
F
0<F<F
0(where F
0is the
solution of Eqs. (4-13) or (4-14), respec-
tively) the thermal equilibrium stateis 
not a pure homogeneous phase: rather, the
minimum free energy state is given by the
double-tangent construction to F(
F) and
this corresponds to a mixed phase state(the 
relative amounts of the coexisting phases
are given by the well known lever rule).
Now it is standard practice, dating back to
van der Waals’ interpretation of this equa-
tion of state for fluids, to interpret the part
of F(
F) in Fig. 4-6 which lies above the
F(
F) given by the double-tangent con-
struction as a metastable state provided 
that 
c
T=(∂
2
F/∂F
2
)
T< 0, whereas states
with 
c
T< 0 are considered as intrinsically
unstable states. As will be seen in Sec. 4.4,
this notion is intrinsically a concept valid
only in mean-field theory, but lacks any
fundamental justification in statistical me-
chanics. Schemes such as those shown in
Fig. 4-6 make sense for a local “coarse-
grained free energy function”only (which
depends on the length scale Lintroduced in
Eq. (4-9)), see Sec. 4.2.2, but not for the
global free energy.
After this digression we return to the
generalization of the Landau expansion,
Eq. (4-10), the case where the order param-
eter has vector or tensor character. How
can we find which kinds of term appear in
the expansion?
Basically, there are two answers to this
question. A general method, which follows
below, is a symmetry classification based
on group theory techniques (Landau and
Lifshitz, 1958; Tolédano and Tolédano,
1987). A very straightforward alternative
approach is possible if we consider a par-
ticular model Hamiltonian 
∫(for a brief
discussion of some of the most useful mod-
els of statistical mechanics for studying
phase transitions, see Sec. 4.3.1). We can
then formulate a microscopic mean-field
approximation (MFA), such as the effec-
tive field approximation of magnetism 
(Smart, 1966) or the Bragg-Williams ap-
proximation for order–disorder phenomena
in alloys (De Fontaine, 1979), where we
then expand the MFA Helmholtz energy di-
rectly.
As an example of this approach, we con-
sider a specific model of a ternary alloy
where each lattice site imay be taken by ei-
ther an A, a B, or a C atom, assuming con-
centrations c
A=c
B=c
C=1/3, and assuming
that an energy Jis achieved if two neigh-
boring sites are taken by the same kind of
atom. This is a special case of the q -state
Potts model (Potts, 1952; Wu, 1982), the
Hamiltonian being
(4-20)
∫
Potts==−…
〈〉
∑
ij
SS i
JS q
ij
d,{,,,}12www.iran-mavad.com
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308. The states 1, 2, and 3 of the Potts spin stand
for the three kinds of atomic species A, B,
and C here and ·ijÒdenotes a sum over
nearest neighbor pairs in the lattice. In the
MFA, we construct the Helmholtz energy
F=U–TSsimply by expressing both en-
thalpy Uand entropy S in terms of the frac-
tions n
aof lattice sites in states a. The en-
tropy is simply the entropy of randomly
mixing these species, and, using Stirlings
equation, this yields the standard expres-
sion (see elementary textbooks on statisti-
cal thermodynamics):
(4-21)
In the enthalpy term, MFA neglects correla-
tions in the occupation probability of neigh-
boring sites. Hence the probability of find-
ing a nearest neighbor pair in 
astate is sim-
ply n
2
a, and in a lattice with coordination
number zthere are z /2 pairs per site. Hence
(4-22)
and thus
(4-23)
We could directly minimize Fwith respect 
to n
a, subject to the constraint 
since each site should be occupied. In order
to make contact with the Landau expan-
sion, however, we rather expand Fin terms
of the two order parameter components
F
1=n
1–1/3 and F
2=n
2–1/3 (note that all
n
i=1/3 in the disordered phase). Hence we
recognize that the model has a two-compo-
nent order parameter and there is no sym-
metry between 
F
iand –F
i. So cubic terms
in the expansion of Fare expected and do
occur, whereas for a properly defined order
parameter, there cannot be any linear term
a
a=
=
1
3
1∑n,
F
Vk T
zJ
kT
nnn
BB ==
=−+ ∑∑
2
1
3
2
1
3a
a
a
aa
ln
U
zJV
n=
=
− ∑
2
1
3
2a
a
SVnn=
=
−∑
a
aa1
3
ln
in the expansion:
As expected, there is a temperature T
0
(=zJ/3k
B) where the coefficient of the
quadratic term changes sign.
Of course, for many phase transitions a
specific model description is not available, and even if a description in terms of a model Hamiltonian is possible, for compli- cated models the approach analogous to Eqs. (4-20) and (4-24) requires tedious cal- culation. Clearly the elegant but abstract Landau approach based on symmetry prin- ciples is preferable when constructing the Landau expansion. This approach starts from the observation that usually the disor- dered phase at high temperatures is more “symmetric” than the ordered phase(s) oc- curring at lower temperature. Recalling  the example described in Fig. 4-3, referring to Fe–Al alloys: in the high temperature A2 phase, all four sublattices a, b, c, and d are completely equivalent. This permuta- tion symmetry among sublattices is broken in the B2 phase (FeAl structure) where the concentration on sublattices a, c differs from the concentration on sublattices b, d. A further symmetry breaking occurs when we go from the B2 phase to the D0
3phase
(Fe
3Al structure), where the concentration
on sublattice b differs from that on sublat- tice d. In such cases the appropriate struc- ture of the Landau expansion for Fin terms
of the order parameter 
Fis found from the
principle that Fmust be invariant against
all symmetry operations of the symmetry group G
0describing the disordered phase.
In the ordered phase, some symmetry ele-
F
Vk T
zJ
kT
zJ
kT
BB
B
=
(4-2 )
−−
+−
⎛
⎝
⎜
⎞
⎠
⎟++
+++…
6
3
31
3
9
2
4
1
2
2
2
12
1
2
212
2
ln
()
()
FFFF
FF FF
256 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

309. 4.2 Phenomenological Concepts 257
ments of G
0fall away (spontaneously
broken symmetry); the remaining symme-
try elements form a subgroup Gof G
0.
Now the invariance of Fmust hold separ-
ately for terms 
F
k
of any order kand this
requirements fixes the character of the
terms that may be present.
Rather than formulating this approach
systematically, which would require a
lengthy and very mathematical exposi-
tion (Tolédano and Tolédano, 1987), we
rather illustrate it with a simple example.
Suppose a cubic crystal exhibits a transi-
tion from a para-electric to a ferroelectric
phase, where a spontaneous polarization
P=(P
1,…,P
n), n= 3, appears. Fis then
given as follows:
Whereas the quadratic term of a general
dielectric medium would involve the in-
verse of the dielectric tensor,
this term is completely isotropic for cubic
crystals. Inversion symmetry requires in-
variance against PÆ–Pand hence no
third-order term occurs. The fourth-order
term now contains the two “cubic invari-
ants” Here we invoke 
the principle that all terms allowed by sym-
metry will actually occur. Now Eq. (4-25)
leads, in the framework of the Landau the-
ory, to a second-order transition if both
u> 0 andu+ u¢> 0 (4-26)
whereas otherwise we have a first-order
transition (then terms of sixth order are
needed in Eq. (4-25) to ensure stability).
PP
i
i
i
i
42 2and∑∑ ()
.
ij
ij i j
PP∑
−
() ,c
el
1
Fx
kT
F
kT
uPu PP
R
d
P
n
j
n
j
n
[()]
()
P
xP
BB
el
==
=
=d
(4-2 )
0 12
1
4
1
22
2
1
2 1
2
1
4
2
5
+
⎡
⎣
⎢
++ ′
⎛
⎝
⎜
⎞
⎠
⎟
+…
+∇+…
⎤
⎦
⎥
∫
∑∑
∑
−
<
c
m
m
m
m
m
m
This approach also carries over to cases
where the order parameter is a tensor. For
example, for elastic phase transitions 
(Cowley, 1976; Folk et al., 1976) the order
parameter is the strain tensor {
e
ik}. Apply-
ing the summation convention (indices oc-
curring twice in an expression are summed
over), the Landau expansion is 
F[
e
ik(x)] 
= F
0+ Údx(–
1
2
c
iklme
ike
lm
+ –
1
3
c
(3)
iklmrs
e
ike
lme
rs
+ –
1
4
c
(4)
iklmrsuv
e
ike
lme
rse
uv
+ … + gradient terms) (4-27)
Here the c
iklmare elastic constants and 
c
(3)
iklmrs
and c
(4)
iklmrsuv
analogous coefficients
of higher order (“anharmonic”) terms. For
most elastic transitions symmetry permits
some nonzero c
(3)
iklmrs
and hence leads to
first-order transitions. Examples of such
systems are the “martensitic transitions” in
Nb
3Sn and V
3Si; there the elastic distortion
jumps at the transition from zero to a very
small value (
e
ik≈10
–4
) and a Landau ex-
pansion makes sense. Note that the Landau
theory is not very useful quantitatively for
transitions which are very strongly first or-
der, since in that case high-order terms in the
Taylor expansion are not negligible. More
about martensitic phase transformations can
be found in the Chapter by Delaey (2001).
Just as in a ferromagnetic transition the
inverse magnetic susceptibility 
c
–1
van-
ishes at T
c(Eq. (4-4)) and in a ferroelectric
transition the inverse dielectric susceptibil-
ity 
c
el
–1vanishes at T
c(Eq. (4-25)), in a
“ferroelastic” transition one of the elastic
constants c
iklmvanishes at T
0. For example,
in KCN such a softening is observed for the
elastic constant c
44. Note that this material
has already been mentioned as an example
of a phase transition where the order pa-
rameter is the electric quadrupole moment
tensor (Eq. (4-8)), describing the orienta-www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

310. tion of the dumbbell-shaped CN
–
-ions: 
at the same time, it can be considered as 
an example for an elastic transition, where
the order parameter is the strain tensor
(
e
mn(x))! Such an ambiguity is typical of
many first-order transitions, because vari-
ous dynamical variables (such as a local
dielectric polarization P(x), a local quadru-
pole moment f
mn(x) and the strain e
mn(x)
may occur simultaneously in a crystal and
are coupled together. In a Landau expan-
sion, such couplings are typically of the bi-
quadratic energy–energy coupling type,
i.e., various types of mixed fourth-order
terms occur, such as P
2
(x) e
mne
m¢n¢u
mn m¢n¢.
Although this is also true for second-order
transitions, there the “primary order pa-
rameter” is distinct from the fact that the
associate inverse response function 
c
T
–1
vanishes at T
c. For the “secondary order
parameters” the corresponding inverse re-
sponse function stays finite. With first-
order transitions, the inverse response
function of the primary order parameter
would vanish at the hypothetical tempera-
ture T
0(stability limit of the disordered
phase), which, however, cannot normally
be reached in a real experiment, see Sec.
4.4. K
+
CN
–
is a good example of such a
complicated situation with three local or-
der parameters coupled together, since the
CN
–
ion also has a dipole moment. After
the first-order transition at T= 110 K from
the cubic “plastic crystal” phase (where
there is no long-range orientational order,
the CN
–
dumbbells can rotate) to the tet-
ragonally distorted, orientationally ordered
phase, a second first-order transition oc-
curs at T= 80 K to an antiferroelectric
phase where the dipole moments order. In
this system, only a microscopic theory 
(Michel and Naudts, 1977, 1978; De Raedt
et al., 1981; Lynden-Bell and Michel, 1994)
could clarify that the elastic interaction
between the quadrupole moments f
mn(x)
(mediated via acoustic phonons) is more
important than their direct electric quadru-
pole–quadrupole interaction. The elastic
character of the phase transition is therefore
an intrinsic phenomenon for this material.
We briefly discuss the appropriate con-
struction of the order parameter (compo-
nents) for order–disorder transitions. Fig.
4-3 (and Eq. (4-7)) illustrated how we can
visualize the ordered structure in real
space, and apply suitable symmetry opera-
tors of the point group. However, it is often
more convenient to carry through a corre-
sponding discussion of the ordering in re-
ciprocal space rather than in real space (re-
member that the ordering shows up in
superlattice Bragg spots appearing in the
reciprocal lattice in addition to the Bragg
spots of the disordered phase). For exam-
ple, consider rare gas monolayers adsorbed
on graphite: at low temperatures and pres-
sures, the adatoms form a ÷
–
3 structure
commensurate with the graphite lattice.
This ÷
–
3 structure can be viewed as a trian-
gular lattice decomposed into three sublat-
tices, such that the adatoms preferentially
occupy one sublattice. Mass density waves
are taken as an order parameter (Bak et al.,
1979; Schick, 1981):
(4-28)
Here the q
aare the three primitive vectors
associated with the reciprocal lattice of the
rare gas monolayer (abeing the lattice
spacing of the triangular lattice)
(4-29)
q
q
q
1
2
3
2
0
1
3
21
2
1
23
21
2
1
23
=
=
=
p
p
p
a
a
a
,
,
,
⎛
⎝
⎜
⎞
⎠
⎟
−−
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟
ry
y
a
a
aa
() [ exp( )
exp( )]
xqx
qx
=i
=1
3
∑ ⋅
+−⋅
−
258 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

311. 4.2 Phenomenological Concepts 259
and y
aand y
–aare the (complex!) order
parameter amplitudes. In constructing the
free energy expansion with the help of Eq.
(4-28), note that the periodicity of the
underlying graphite lattice allows invariant
“umklapp” terms (the phase factors of
third-order terms add up to a reciprocal lat-
tice vector of the graphite lattice). Keeping
only those terms in the Landau expansion,
and which are nonzero in the ÷
–
3 structure,
with the real order parameter components
(i=÷
––
–1)
(4-30)
the expression
is obtained. This is equivalent to the result
for the three-state Potts model, Eq. (4-24)
(Alexander, 1975), after the quadratic form
is diagonalized. A three-dimensional ana-
log of this order (where the planes exhibit-
ing ÷
–
3 structure are stacked together to
form a hexagonal lattice) occurs in the
intercalated compound C
6Li (Guerard and
Herold, 1975; Bak and Domany, 1979).
Similarly to the description of the den-
sity modulation in the superstructure of 
adsorbed layers (in two dimensions) or
interstitial compounds (in three dimen-
sions) in terms of mass density waves of
the adsorbate (or interstitial, respectively),
the superstructure ordering in binary alloys
can be described in terms of concentration
waves (Khachaturyan, 1962, 1963, 1973,
1983; De Fontaine, 1975, 1979). The same
concept was used even earlier to describe
the magnetic ordering of helimagnetic spin
F
Vk T
r
wu
B
= (4-31)−+
+− ++
1
2
1
3
3
1
4
1
2
2
2
1
3
12
2
1
2
2
22()
()()FF
FFF FF
F
F
1
1
3
2
1
3 23
23
=
=i
=
=a
aa
a
aa
yy
yy∑
∑ −
−
−
−( )/( )
( )/( )
structures in terms of „spin density waves“
(Villain, 1959; Kaplan, 1959). We shall
outline the connection between this ap-
proach and the MFA, generalizing the ap-
proach of Eqs. (4-21)–(4-23) slightly, in
Sec. 4.3. Here we only mention that these
concepts are closely related to the descrip-
tion of structural transitions in solids,
where the order parameter can often be
considered as a “frozen phonon”, i.e., a dis-
placement vector wave (Bruce and Cowley,
1981). Note also that the approach of con-
centration waves is not restricted to solids,
but can also be used to describe meso-
phases in fluid block-copolymer melts
(Leibler, 1980; Fredrickson and Helfand,
1987; Fredrickson and Binder, 1989; Bin-
der, 1994), see Fig. 4-7, in liquid crystal-
line polymers, etc.
As a final remark in this section we men-
tion that not for all phase transitions in sol-
idsthere does exist a group–subgroup rela-
tionship between the two groups G
1and G
2
describing the symmetry of the phases co-
existing at the transition. These phases can-
not be distinguished by an order parameter
which is zero in one phase and becomes
nonzero in the other. Such transitions must
be of first order. Examples of this situation
are well known for structural phase transi-
tions, e.g., the tetragonal–orthorhombic
transition of BaTiO
3or the “reconstruc-
tive” transition from calcite to aragonite
(Guymont, 1981). For the so-called “non-
disruptive transitions” (Guymont, 1981),
the new structure can still be described in
the framework of the old structure (i.e., its
symmetry elements can be specified, the
Wyckoff positions can be located, etc.).
Landau-type symmetry arguments still
yield information on the domain structures
arising in such phase transitions (Guymont,
1978, 1981). The tetragonal–orthorhombic
transition of BaTiO
3is considered to be an
example of such a non-disruptive transi-www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

312. tion, while “reconstructive transitions” and
martensitic transformations of the type of
the f.c.c.–b.c.t. transition in Fe are “disrup-
tive”(see the Chapter by Delaey (2001)).
We shall not go into detail about such prob-
lems here, however.
4.2.2 Second-Order Transitions
and Concepts about Critical Phenomena
(Critical Exponents, Scaling Laws, etc.)
We now return to the case where at a
critical point the order parameter of a sec-
ond-order transition vanishes continuously
(Fig. 4-1). We consider the accompanying
critical singularities (Fig. 4-2), in the
framework of Landau theory (Eq. (4-10)).
For a proper understanding of the critical
fluctuations of the order parameter, we
must no longer restrict the treatment to the
homogeneous case —
F(x)∫0 as was done
in Eqs. (4-12) and (4-13); we now wish to
consider, for example, the response to an
inhomogeneous, wavevector-dependent or-
dering field H(x):
H(x) = H
qexp (iq∙x) (4-32)
Although for a magnetic transition such a
field cannot be directly applied in the la-
boratory, the action of such fields can be
indirectly probed by appropriate scattering
measurements, e.g., magnetic neutron scat-
tering. This is because the scattering can 
be viewed as being due to the inhomogene-
ous (dipolar) field that the magnetic mo-
ment of the neutron exerts on the probe.
Therefore, the scattering intensity for a
scattering vector qis related to the wave-
vector-dependent susceptibility (see Kittel
(1967)). Similarly, by the scattering of 
X-rays the wavevector-dependent response
function to the sublattice ordering field can
be measured in ordering alloys such as b-
CuZn (see Als-Nielsen (1976) for a discus-
sion of such scattering experiments in vari-
ous systems).
Hence in order to deal with Eq. (4-10)
we now have to minimize a functional
rather than a simple function as was done
in Eq. (4-12). In order to do so, we note
that the problem of minimizing Údxf(
y,
—
y) in Eq. (4-10) is analogous to the prob-
lem of minimizing the action in classical
mechanics, W=Údt
(x,x˙), where is the
Lagrangian and x˙=dx/dt, the velocity of 
a particle at point x(t= time). Just as in
classical mechanics, where this problem 
260 4 Statistical Theories of Phase Transitions
Figure 4-7.(a) Chemical architecture of a diblock
copolymer. A diblock copolymer consists of a poly-
merized sequence of A monomers (A-block) cova-
lently attached to a similar sequence of B monomers.
(b) The microphase separation transition occurs
when a compositionally disordered melt of copoly-
mers, which are in random coil configurations (right)
transforms to a spatially periodic, compositionally
inhomogeneous phase (left) on lowering the temper-
ature. For nearly symmetric copolymers the ordered
phase has the lamellar structure shown. Since the
wavelength of the concentration wave here is of the
order of the coil gyration radius, which may be of the
order of 100 Å for high molecular weight, the order-
ing occurs on a “mesoscopic” rather than micro-
scopic scale (Å) and hence such phases are called
“mesophases”. From Fredrickson and Binder (1989).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

313. 4.2 Phenomenological Concepts 261
is solved in terms of the Euler–Lagrange
equation
we conclude here that
in order that the Helmholtz energy func-
tional is a minimum. Eq. (4-10) thus yields
(4-33)
Using
F(x)=F
0+DF(x)=F
0+DF
qexp(iq∙x)
Eqs. (4-32) and (4-33) yield, on linearizing
in D
F
qfor small H
q, the wavevector-de-
pendent susceptibility
From Eq. (4-34) we can simply read off 
the temperature dependence of the suscep-
tibility 
c
T(Eq. (4-4)) and the correlation
length 
xof order parameter fluctuations,
which is generally expected to behave as
(4-35)
Above T
c, the terms 3u F
0
2+5vF
0
4=0,
hence
c
x
T
kTr rT T
Rrd RrdTT
= = (4-3 )
==
Bc
c
11
6
21 2
′−
′ −
−
()
/( ) ( / ) ( )
/
x
x
x
n
n
=
cc
cc
ˆ
(/ ) ,
ˆ
(/),
TT T T
TT T T
−>
′−<
⎧
⎨
⎩ −
−′
1
1
c
c
x()q
q
q
≡
+++
⎡
⎣
⎢
⎤
⎦
⎥
+
−
DF
FF
H
kT
ru
R
d
q
q
T
=
=(4-3)
B
1
35
1
4
0
2
0
4
2
2
1
22v
ru
R
d
H
kTFF F
F() () ()
()
()
xxx
x
x
++
−∇
35
2
2
v
=
B
(/ ) [/( )]∂∂ −∇∂∂∇ffyy =0
(/) [/(/)]∂∂− ∂∂≡≡x
t
xt
d
d
dd =0
For T<T
c, using Eq. (4-13) the term 5v F
0
4
near T
cis still negligible, and hence
Eqs. (4-36) and (4-37) simply imply the
well known Curie–Weiss law, i.e., the ex-
ponents 
g, g¢, defined in Eq. (4-4), are, in
the framework of the Landau theory,
g= g¢= 1 (4-38)
Comparing Eqs. (4-35) and (4-37) we also
conclude that
n= n¢= 1/2 (4-39)
The physical meaning of the correlation
length 
xis easily recognized if the well
known fluctuation relationship
(4-40)
is expanded to second order in qas
which can be written as
Comparing Eqs. (4-34) and (4-42), we find
that in terms of the order parameter corre-
lation function G(x),
(4-43)
G() () ()xx=〈〉−FF F0
0
2
c() [ () () ]
[()() ]
[()() ]
qx
x
x
x
x
x
=
(4-42)
B
1
0
1
2
0
0
0
2
2
2
0
2
0
2
kT
q
d
x
∑
∑
∑〈〉−
×−
⎧
⎨
⎩
〈〉−
〈〉−
⎫
⎬
⎭FF F
FF F
FF F
c() [ () () ]
()[()() ]
qx
qx x
x
x
≈
⎧
⎨
⎩
〈〉−
−⋅〈 〉−
⎫
⎬
⎭ ∑
∑
1
0
1
2
0
0
2
2
0
2
kT
B
 (4-41)
FF F
FF F
c( ) exp( )
[()() ]
qqx
x
x
=i
B
1
0
0
2
kT
∑ ⋅
×〈 〉−
FF F
c
x
T
kTr rT T
RrdRrdTT
= = (4-3 )
==
Bc
c
−
′−
− ′ −
−
1
2
1
2
7
22
21 2
()
/( ) ( / ) ( )
/www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

314. and c
Tand x
2
are related to the zeroth and
second moment of the correlation function:
(4-44)
(4-45)
Eq. (4-44) clearly shows that the critical di-
vergence of 
c
Toccurs because the correla-
tions G(x) become long ranged: whereas
off T
cthe correlation function for large |x|
decays exponentially,
lnG(x) Æ–|x|/
x,|x|Æ• (4-46)
at T
cwe have a power-law decay of the cor-
relation function:
G(x) = Gˆ|x|
–(d–2+h)
,T= T
c (4-47)
The exponent describing this critical decay
has been defined such that
h= 0 (4-48)
in the Landau theory, whereas in general
h∫0, as will be discussed below.
In order to make our collection of critical
exponents complete, we also consider the
critical isotherm
F
0= DˆH
1/d
,T= T
c (4-49)
Using Eq. (4-33) for a uniform field we
conclude that
F
0= (k
BTu)
–1/3
H
1/3
(4-50)
i.e., Dˆ=(k
BTu)
–1/3
and
d= 1/3 (4-51)
in the framework of the Landau theory. 
Finally, we turn to the specific heat, Eq. 
(4-5), which behaves rather pathologically
in the Landau theory: for T>T
cand H=0,
Eq. (4-10) just implies F=F
0, i.e., the 
specific heat associated with the ordering
described by
Fis identically zero above
x
221
2
=
d
xG G
xx
xx∑∑() ()
c
T
kT
G=
B
1
x
x∑()
T
c. For T<T
c, we have instead, from Eqs.
(4-10) and (4-13)
(4-52)
which implies C=r¢
2
(T/T
c)/(4k
Bu) for
T<T
c. This jump singularity of the specific
heat at T
cinstead of the power law in Eq.
(4-5) is formally associated with vanishing critical exponents:
a= a¢= 0 (4-53)
The question must now be asked whether this description of critical phenomena is accurate in terms of the Landau theory. The free energy functional F[
F(x)] in Eq. 
(4-10) should be considered as an effective Hamiltonian from which a partition func- tion Zcan be obtained so that the true
Helmholtz energy becomes:
F=–k
BTlnZ (4-54)
=–k
BTlnÚd[F(x)] exp {–F[ F(x)]/k
BT}
It is natural to expect that the main contri-
bution to the functional integral comes
from the region where the integrand is larg-
est, i.e., the vicinity of the point where
F[
F(x)]/k
BThas its minimum. If we as-
sume that the distribution over which the
average in Eq. (4-54) is a delta function at
the value 
F
0yielding the minimum, i.e.,
we assume that fluctuations of the order
parameter make a negligible contribution
to the functional integral, then the mini-
mum of F is equivalent to the minimum of
F(
F). In general, however, this is not true,
as the effects of fluctuations modify the
critical behavior drastically, and the Lan-
dau theory does not hold for systems such
as b-CuZn.
A first hint of the conditions for which
the Landau theory is valid can be obtained
FF
kTV
ru
r
u
r
TT
ku
−
+
⎛
⎝
⎞
⎠
−
≈−′
−
0
0
2
0
2
2
2
2
2
24
4
1
4
B
c
B
=
=FF
(/)
262 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

315. 4.2 Phenomenological Concepts 263
by considering the effect of fluctuations
within the context of the Landau theory it-
self. The criterion Ginzburg (1960) sug-
gests is that the Landau theory is valid if
the mean-square fluctuation of the order
parameter in a correlation volume is small
in comparison with the order parameter
square itself:
·[
F(x) –F
0]
2
Ò
L=xOF
0
2 (4-55)
Here Lis the coarse-graining lengthintro-
duced in Eq. (4-9) in order to transform the
microscopic lattice description of ordering
phenomena in solids to a continuum de-
scription. On the lattice level, the local or-
der parameter 
F
ishows a rapid variation
from one lattice site to the next, and the
mean-square fluctuation is very large. For
example, for an Ising model of an aniso-
tropic ferromagnet,
(4-56)
Where Jis the exchange interaction, the
sum ·i,jÒextends once over all nearest
neighbor pairs, and S
ipoints in the direc-
tion of the local magnetic moment at lattice
site i, we have 
F
i=S
i, i.e., F
i
2∫1. In this
case ·[
F
i–·F
iÒ]
2
Ò=1–M
s
2is never small
in comparison with the square of the spon-
taneous magnetization M
s
2near T
c. Carry-
ing out the local averaging defined in Eq.
(4-9) reduces this local fluctuation in a
“volume” L
d
. This averaging should, in
principle, be carried out over a length scale
Lthat is much larger than the lattice spac-
ing abut much smaller than the correlation
length 
x,
aOLO
x (4-57)
Whereas for a study of critical phenomena
it is permissible to average out local effects
on the scale of a lattice spacing, relevant
spatial variations do occur on the scale of
x, as we have seen in Eqs. (4-34) and 
∫
Ising==−− ±
〈〉
∑∑JSSHSS
ij
ij
i
ii
,
,1
(4-46). Replacing the right inequality in Eq. (4-57) by the equality, L =
x, as done in
Eq. (4-55), gives therefore the maximum permissible choice for L, for which the
fluctuations in Eq. (4-55) are smallest. Even then, however, it turns out that Eq. (4-55) typically is not fulfilled for Tclose to T
c. In
order to see this, we write Eq. (4-55) in terms of the local variable 
F
iand denote
the number of lattice sites icontained 
L
d
for the choice L= xas N(t), where
t∫1–T/T
c. Then Eq. (4-55) becomes
(4-58)
where we have denoted the volume of size L=
xcentered at xas V
x(x). Making use of
the translational invariance of correlation functions and ·
F
iÒ=M
s(t), the spontane-
ous order parameter which is independent of i, Eq. (4-58) becomes 
(4-59)
If the sum over correlations in Eq. (4-59) were to extend over all space, it would sim- ply be the “susceptibility” k
BTc(t) (see
Eq. (4-42)). Since the sum does contain just the volume region over which the 
F
is
are strongly correlated with each other, the sum clearly is of the order of fk
BTc(T),
where f<1 is a factor of order unity, which
should have no critical (vanishing or diver- gent) temperature dependence as TÆT
c.
Hence in the inequality Eq. (4-59) this fac- tor may also be omitted, and we conclude (Als-Nielsen and Laursen, 1980) that
c(t) ON(t) OM
s
2(t) (4-60a)
Making use of N(t)=[
x(t)/a]
d
, abeing the
lattice spacing, we obtain:
const. O[
x(t)]
d
M
s
2(t)c
–1
(t) (4-60b)
Nt M t
NtMt
iV
ii
() [ ()]
() ()
()Œ
x
x
∑〈〉−FF
=s
s
0
2
22
O
[
−〈 〉
]
∑∑
iV
ii
iV
i
ŒŒ
xx() ()xx
FF F
2 2
Owww.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

316. where in suitable units the constant on the
left-hand side of the equality is of order
unity. Using Eqs. (4-4), (4-6), and (4-35)
we obtain (anticipating 
g= g¢and n= n¢)
const. Ot
–nd+2b+g
(4-61)
Inserting the Landau values for the critical
exponents 
b= 1/2, n= 1/2, g= 1 and know-
ing the critical amplitude of 
x~R, the inter-
action range, Eq. (4-60) implies
const. O(R/a)
d
t
(d–4)/2
(4-62)
This condition for the validity of the Lan-
dau theory for d< 4 always breaks down 
as tÆ0 (TÆT
c). In fact, for d< 4 close
enough to T
ca regime occurs where fluctu-
ations dominate the functional integral (Eq.
(4-54)). The “crossover” from the mean-
field regime (where the Landau description
is essentially appropriate) to the non-mean-
field regime occurs at a reduced tempera-
ture distance t=t
cr, where Eq. (4-62) is
treated as an equality: for d= 3, 
t
cr~(R/a)
–6
(4-63)
Hence systems with a large but finite range
of interaction behave in an essentially Lan-
dau-like manner. An example of such a be-
havior is the unmixing critical point in
polymer fluid mixtures with a high degree
of polymerization N: since each coil has a
radius r
c~N
1/2
, the monomer density in-
side the sphere taken by a polymer coil is
only of order 
r~N/r
c
3~N
–1/2
, which im-
plies that each coil interacts with N
1/2
neighbor coils, and the effective interaction
volume (R /a)
d
in Eqs. (4-62) and (4-63)
should be taken as N
1/2
, i.e., t
cr~N
–1
(De
Gennes, 1979; Binder 1984c, 1994).
At this point, we emphasize that using
N(t)=[
x(t)/a]
d
in Eq. (4-60) is not valid in
systems with long-range anisotropic inter-
actions, such as uniaxial dipolar magnets.
For a system such as LiTbF
4(Als-Nielsen
and Laursen, 1980), the essential part of
the magnetic Hamiltonian is a magneto-
static dipole–dipole interaction,
(4-64)
where 
mis the magnetic moment per spin
and zthe coordinate of xin the direction of
the uniaxial anisotropy. Unlike the iso-
tropic Ising Hamiltonian in Eq. (4-56), here
the sign of the interaction depends on the
direction in the lattice. Therefore, fluctua-
tions display an essential anisotropy: in-
stead of the isotropic result (Eq. (4-34)) of
the well known Ornstein-Zernike type, 
the wavevector-dependent susceptibility
c(q)=S(q)/k
BT(S(q) = structure factor)
becomes anisotropic:
S(q) ~ [
x
–2
+q
2
+g(q
z/q)
2
]
–1
(4-65a)
where q
zis the z-component of qand gis a
constant. Whereas Eq. (4-32) implies that
S(q
x=1)=S(q= 0)/2, the equation q x=1
which for Eq. (4-32) defines a sphere of ra-
dius 
x
–1
in q-space now defines an aniso-
tropic surface. From
x
–2
=q
2
+g(q
z/q)
2
(4-65b)
we obtain an object having the shape of a
flat disc, with radius 1/
xin the q
x–q
yplane
but maximum extension of order 1/
x
2
in
the q
zdirection. This object, defined by
S(q)=–
1
2
max {S (q)}, can be interpreted as
the Fourier transform of the correlation
volume. This implies that the correlation
volume in real space is a long ellipsoid,
with linear dimensions 
xin the q
x,q
ydi-
rection but with linear dimension 
x
2
in the
q
zdirection. In d-dimensions, this argu-
ment suggests for uniaxial dipolar systems:
N(t)=[
x(t)/a]
d+1
(4-66)
which yields in Eq. (4-60)
const. Ot
–n(d+1)+2b+g
=t
–(d–3)/2
(4-67)
∫==−
−
±
≠
∑m
2
22
53
1
ij
ij i
z
SS S
x
x||
,
264 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

317. 4.2 Phenomenological Concepts 265
where in the last equality, the Landau ex-
ponents have been used. Comparing Eqs.
(4-61) and (4-67) shows that d-dimensional
uniaxial dipolar systems somehow corre-
spond to (d +1)-dimensional systems with
isotropic short-range forces. Therefore the
marginal dimension d*, above which the
Landau theory predicts the values of criti-
cal exponents correctly, is d*= 3 for uniax-
ial dipolar systems, unlike the standard iso-
tropic short-range case where d*=4.
For elastic phase transitions in cubic
crystals where the combination of elastic
constants c
11–c
12softens as the critical
point is appproached, such as the system
PrAlO
3, the structure factor S(q) has the
form:
(4-68)
S(q) ~ [
x
–2
+q
2
+A(q
z/q)
2
+B(q
^/q)
2
]
–1
where q=(q
z,q
^) and A and B are con-
stants. In this case the correlation vol-
ume defined from the condition S(q)=
–
1
2
max {S(q)} in real space is a flat disk
with radius 
x
2
and diameter x, i.e., for elas-
tic systems:
N(t) = [
x(t)/a]
d+2
(4-69)
In this case, the Ginzburg criterion, Eq. 
(4-60), yields
const. Ot
–n(d+2)+ 2b+g
=t
–(d–2)/2
(4-70)
instead of Eqs. (4-62) or (4-67), respec-
tively. Since the marginal dimensionality
for such systems is d*= 2, three-dimen-
sional systems should be accurately de-
scribed by the Landau theory, and this is
what is found experimentally for PrAlO
3.
Examples for elastic phase transitions
which are of second order are scarce, as is
understandable from the symmetry consid-
erations which lead to first-order transi-
tions in most cases (Cowley, 1976; Folk
et al., 1976). However, for many first-order
ferroelastic transitions the Landau descrip-
tion also fits the experimental data very
well over a wide temperature range (Salje,
1990).
For all other systems we do not expect
the Landau critical exponents to be an ac-
curate description of the critical singular-
ities. This is borne out well by the exact so-
lution for the two-dimensional Ising model
(Onsager, 1944; McCoy and Wu, 1973),
where the critical exponents have the values
a= 0 (log),b=1/8,g= 7/4,
d= 15,n=1,h= 1/4 (4-71)
Obviously, these numbers are a long way
from the predictions of the Landau theory.
Note that 
a= 0 in Eq. (4-71) has the mean-
ing C~|log|t||, unlike the jump singular-
ity of the Landau theory.
Even in the marginal case of uniaxial di-
polar ferromagnets (or uniaxial ferro-
electrics; see, e.g., Binder et al. (1976) for
a discussion), the Landau theory is not
completely correct; it turns out (Larkin and
Khmelnitskii, 1969; Aharony, 1976) that
the power laws have the Landau form but
are modified with logarithmic correction
factors:
c
T=G
ˆ
±t
–1
|lnt|
1/3
, tÆ0 (4-72a)
C=Aˆ
±|lnt|
1/3
, tÆ0 (4-72b)
M
s=Bˆ(–t)
1/2
|ln(–t)|
1/3
,tÆ0 (4-72c)
M
s|
T
c
=DˆH
1/3
|lnH|
1/3
,HÆ0 (4-72d)
Although for uniaxial ferroelectrics the ex-
perimental evidence in favor of Eq. (4-72)
is still scarce, convincing experimental ev-
idence does exist for dipolar ferromagnets
such as LiTbF
4(see, e.g., Als-Nielsen and
Laursen (1980) for a review).
Eq. (4-72) results from including fluctu-
ation contributions to the functional inte-
gral in Eq. (4-54) systematically, which can
be done in a most powerful way be renor-
malization group theory (Aharony, 1973,
1976; Fisher, 1974; Wilson and Kogut,www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

318. 1974; Ma, 1976; Amit, 1984; Yeomans,
1992). A description of this theory is be-
yond the scope of this chapter; we only
mention that it is this method which yields
highly accurate approximations for the val-
ues of critical exponents of three-dimen-
sional systems. For example, for the uniax-
ial magnets (as described by the Ising Ha-
miltonian, Eq. (4-56), experimental exam-
ples being the antiferomagnets MnF
2or
FeF
2) or the ordering alloy b-brass, the ex-
ponents are predicted to be (LeGuillou and
Zinn-Justin, 1980):
a≈0.110,b≈0.325,g≈1.240,
d≈4.82,n≈0.63,h≈0.032 (4-73)
Whereas within the Landau theory the “or-
der parameter dimensionality” n does not
matter, as far as the values of the critical
exponents are concerned, n does matter if
we go beyond the Landau theory. For ex-
ample, for isotropic magnets as described
by the well known Heisenberg model of
magnetism,
(4-74)
S
ibeing a unit vector in the direction of the
magnetic moment at lattice site i, we have
(LeGuillou and Zinn-Justin, 1980):
a≈– 0.116,b≈0.365,g≈1.391,
d≈4.82,n≈0.707,h≈0.034 (4-75)
Again these results are in fair agreement
with available experimental data, such as
the isotropic antiferromagnet RbMnF
3(see
Als-Nielsen (1976) for a review of experi-
ments on critical point phenomena). Note
that the negative value of 
ain Eq. (4-75)
implies that the specific heat has a cusp of
finite height at T
c.
The renormalization group theory also
provides a unifying framework and justifi-
cation for two important concepts about
critical phenomena, namely “scaling” and
∫
Heis=−⋅−
≠
∑∑
ij
ij i j i
z
JHSSS m
“universality”. Again we wish to convey only the flavor of the idea to the reader, rather than to present a thorough discus- sion. Consider, for example, the decay of the correlation function of order parameter fluctuations at the critical point, Eq. (4-47): changing the length scale from rto r¢=
Lr
would change the prefactor Gˆbut leave the
power-law invariant. The physical inter- pretation of this fact is the “fractal struc- ture” of critical correlations (Mandelbrot, 1982): just as a fractal geometric object looks the same on every length scale, the pattern of critical fluctuations looks the same, irrespective of the scale. Slightly off T
c, then, there should be only one relevant
length scale in the problem, the correlation length 
x, which diverges as TÆT
c, and de-
tails such as the precise behavior of the correlation functions for interatomic dis- tances should not matter. As a result, it is plausible that the correlation function G(x,
x) should depend only on the ratio of
the two lengths xand 
x, apart from a scale
factor:
G(x,
x)=Gˆx
–(d–2+ h)
G
˜
(x/x) (4-76)
Obviously, the condition G
˜
(0) =1 for the
“scaling function” G
˜
(x/
x) ensures that Eq.
(4-76) crosses over smoothly to Eq. (4-47)
as TÆT
c, where xÆ∞.
From the “scaling hypothesis” in Eq. 
(4-76) we immediately derive a “scaling
law” relating the exponents 
g, nand h. Us-
ing Eq. (4-44) and denoting the surface of 
a d-dimensional unit sphere as U
d(U
d=4p
in d= 3) we obtain
cx
x
x
h
hh
T
x
d
d
kT
Gx
G
kT
Gx
UG
kT
xGx x
UG
kT
G
=d
=d
= d (4-77)
BB c
Bc
Bc
1
1
0
21
0
∑ ∫
∫
∫ ≈
−
∞
−−
∞
()
˜
(, )
ˆ
˜
(/)
ˆ
˜
()
x
′′′
266 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

319. 4.2 Phenomenological Concepts 267
Since the integral over ′=x/ xin Eq. (4-77)
yields a constant, we conclude
c
T~x
(2 –h)
~|t|
–n(2–h)
~|t|
–g
cf. Eqs. (4-4) and (4-35). Thus the scaling
relation results in
g=n(2 –h) (4-78)
Eq. (4-76) indicates that G(x,
x) depends
on the two variables xand 
xin a rather spe-
cial form, namely, it is a homogeneous func-
tion. A similar homogeneity assumption is
also true for the singular part of the free 
energy,
(4-79)
F=F
reg+t
2–a
F
˜
(H
˜
),H
˜
=H G
ˆ
t–g–b
/Bˆ
where the singular temperature dependence
for H
˜
= 0 is chosen to be compatible with
Eq. (4-5), F
regis a background term which
is analytic in both Tand Heven for T =T
c,
while F
˜
(H
˜
) is another “scaling function”.
At this point, we present Eq. (4-79) as a
postulate, but it should be emphasized that
both Eqs. (4-76) and (4-79) can be justified
from Eq. (4-54) by the renormalization
group approach.
Combining Eqs. (4-1) and (4-79) we ob-
tain:
(4-80)
Defining – (
G
ˆ
/Bˆ)F
˜
¢(H
˜
)∫BˆM˜(H
˜
) with 
M˜(0) =1, we obtain:
F=Bˆt
b
M˜(H
˜
), b=2–a–g–b(4-81)
since for H
˜
= 0, Eq. (4-80) must reduce to
Eq. (4-6). Taking one more derivative we
find
c
T=(∂F/∂H)
T=G
ˆ
t–g
M˜¢(H
˜
) (4-82)
hence Eq. (4-79) is compatible with Eq. 
(4-4), as it should be. The condition that
Eq. (4-81) reduces to the critical isotherm
for tÆ0 requires that M˜(H
˜
Æ•)~
H
˜
b/(g+b)
, in order that the powers of t
F
G==−∂ ∂ − ′
−−−
(/ )
ˆ
ˆ
˜
(
˜
)
()
FH
B
tFH
T
2
agb
cancel out. On the other hand, this also yields
F|
T=T
c
~H
1/(1+g/b)
=H
1/d
,
i.e.,
d=1+g/b (4-83)
The scaling assumptions also imply that exponents above and below T
care equal,
i.e, 
a=a¢, g=g¢, and n=n¢. Now there is
still another scaling law which results from considering all degrees of freedom inside a correlation volume 
x
d
to be highly corre-
lated with each other, while different corre- lation volumes can be considered as essen- tially independent: this argument suggests that the singular part of the Helmholtz en- ergy per degree of freedom can be written as F
sing≈[1/N
s(t)]¥const., since indepen-
dent degrees of freedom do not contribute any singular free energy. Since
N
s(t) ~ x
d
~ |t|
–dn
we conclude, by comparison with Eq. 
(4-79), that
d
n= 2 –a (4-84)
Note that this so-called “hyperscaling rela-
tion” with the Landau-theory exponents is
only true at the marginal dimension d*=4,
whereas the other scaling relationships ob-
viously are fulfilled by Landau exponents
independent of the system dimensionality.
However, all scaling relationships (includ-
ing Eq. (4-84)) are satisfied for the two-
and three-dimensional Ising model (Eqs.
(4-71) and (4-73)) and the three-dimen-
sional Heisenberg model, Eq. (4-75).
Let us consider the two-dimensional
Heisenberg model. For this model the ef-
fects of statistical fluctuations are so strong
that they destabilize the ordering alto-
gether: no spontaneous ordering exists in
d= 2 for any system with order parameter
dimensionality n≥2, and a critical point
occurs only at zero temperature, T
c= 0, forwww.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

320. n> 2. Thus d= 2 is the “lower critical di-
mensionality” d
1for n≥2, whereas for
n=1 we have d
1=1: quasi-one-dimensional
orderings are always unstable. The case
n= 2, d= 2 is very special: a phase transi-
tion still occurs at a nonzero T
c, the so-
called “Kosterlitz–Thouless” transition
(Kosterlitz and Thouless, 1973); for T<T
c
we have M
s(T)∫0 while at the same time
x(T)=•, and the correlation function
shows an algebraic decay, Eq. (4-47), with
a temperature-dependent exponent 
h. At
first sight, this behavior may appear fairly
esoteric, but in fact it is closely related 
to the “roughening transition” of crystal
surfaces (Weeks, 1980). Such roughening
transitions have been observed for high-in-
dex crystal faces of various metals (Sala-
non et al., 1988). The power-law decay of
correlations, Eq. (4-47), can be related to
the behavior of the height–height correla-
tion function of a crystal surface in the
rough state (Weeks, 1980). As is well
known, the roughness or flatness of crystal
surfaces has a profound effect on their ad-
sorption behavior, crystal growth kinetics,
etc. (Müller-Krumbhaar, 1977).
The other important concept about criti-
cal phenomena is “universality”: since the
only important length scale near a critical
point is provided by the correlation length
which is much larger than all “micro-
scopic” lengths such as lattice spacing and
interaction range, it is plausible that “de-
tails” on the atomic scale do not matter, and
systems near a critical point behave in the
same way provided that they fall in the
same “universality class”. It turns out that
universality classes (for systems with
short-range interactions!) are determined
by both spatial dimensionality d and order
parameter dimensionality n, and in addi-
tion, the symmetry properties of the prob-
lem: e.g., Eq. (4-25) for n= 3 and u ¢=0
falls in the same class as the Heisenberg
model of magnetism, Eq. (4-74), since the
ferroelectric ordering would also be truly
isotropic. However, in the presence of a
nonvanishing “cubic anisotropy”, u¢≠0, in
general a different universality class re-
sults. The effect of such higher order invar-
iants in the Landau expansion are particu-
larly drastic again in systems with reduced
dimensionality; e.g., the model with n=2
but cubic anisotropy no longer exhibits a
Kosterlitz–Thouless transition, but rather 
a nonzero order parameter is stabilized
again. Owing to the “marginal” character
of the cubic anisotropy in d= 2, however,
the behavior is not Ising-like but rather the
pathological case of a “universality class”
with “nonuniversal” critical exponents oc-
curs. The latter then do depend on “micro-
scopic” details, such as the ratio of the
interaction strengths between nearest and
next nearest neighbors in the lattice (Krin-
sky and Mukamel, 1977; Domany et al.,
1978; Swendsen and Krinsky, 1979; Lan-
dau and Binder, 1985). Again such prob-
lems are not purely academic, but relevant
for order–disorder transitions in chemi-
sorbed layers on metal surfaces, such as O
on W (110) (see Binder and Landau (1989)
for a review of the theoretical modeling of
such systems).
One important consequence of univer-
sality is that all gas–fluid critical points in
three-dimensional systems, all critical
points associated with unmixing of fluid or
solid binary mixtures, and all order–disor-
der critical points involving a single-com-
ponent order parameter (such as b-brass)
belong to the same universality class as the
three-dimensional Ising model. All these
systems not only have the same critical ex-
ponents, but also the scaling functions
M
˜
(H
˜
), G
˜
(′) etc. are all universal. Also,
critical amplitude ratios are universal, 
such as Cˆ/Cˆ
+
, (Eq. (4-4), Aˆ/Aˆ¢(Eq. (4-5)),
or more complicated quantities such as
268 4 Statistical Theories of Phase Transitionswww.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

321. 4.2 Phenomenological Concepts 269
DˆGˆBˆ
d–1
. The theoretical calculation and
experimental estimation of such critical
amplitude ratios are discussed by Privman
et al. (1991).
An interesting problem concerns the
critical singularities associated with nonor-
dering “fields”. These singularities occur
because the ordering field 
F(x) considered
only in Eqs. (4-10) and (4-54) couples to
another quantity: e.g., for an antiferromag-
net in a magnetic field the order parameter
F(the sublattice magnetization) couples to
the uniform magnetization M, to the lattice
parameters, to the electron density, etc.
Since the Hamiltonian is invariant against
interchange of the sublattices which im-
plies a sign change of 
F, this coupling is
quadratic in 
F
2
. Whereas in mean field
theory the resulting critical behavior of the
nonordering “field” is then proportional to
·
FÒ
2
~t
2b
, taking fluctuations into account,
the actual critical behavior is proportional
to ·
F
2
Ò~t
(1–a)
, i.e., an energy-like singu-
larity. Such energy-like singularities are
predicted for the electrical resistivity 
r
elat
various phase transitions (Fisher and
Langer, 1968; Binder and Stauffer, 1976a),
for the refractive index n
r(Gehring and
Gehring, 1975; Gehring, 1977), etc. Look-
ing for anomalies in the temperature deriv-
ative d
r
el(T)/dT, dn
r(T)/dTetc., which
should exhibit specific-heat-like singular-
ities, is often a more convenient tool for lo-
cating such phase transitions than measure-
ments of the specific heat itself.
An important phenomenon occurs when
a nonordering field which is coupled to the
order parameter in this way, such as the
magnetization in an antiferromagnet in an
external field, is held fixed. Suppose we
first study the approach to criticality by let-
ting the uniform magnetic field htend to its
critical value h
c(T): we then have the law
F~[h
c(T)–h]
b
for the order parameter
and M–M
c(T)~[h
c(T)–h]
1–a
for the mag-
netization (due to the coupling 
F
2
(x)M(x)
in the Hamiltonian, as mentioned above).
However, if we now consider the variation
of 
Fwith M, combining both laws we get
(assuming 
a>0)
F~ [M
c(T) – M]
b/(1–a)
(4-85)
Considering now the phase transition in the
space of {M,T} as independent thermody-
namic parameters, we then find that on
crossing the critical line M
c(T) the expo-
nent describing the vanishing of the order
parameter is 
b/(1–a) rather than b. This
effect generally occurs when the critical
line depends on extensive (rather than in-
tensive) thermodynamic variables and is
called “Fisher renormalization” (Fisher,
1968). This is very common for order–dis-
order phenomena in adsorbed layers at
fixed coverage, or in alloys at fixed con-
centration, for unmixing critical points in
ternary mixtures, etc. Other exponents also
become “renormalized” similarly, e.g., 
gis
replaced by 
g/(1–a), etc. A detailed anal-
ysis (Fisher, 1968) shows that no such
“renormalization” of critical exponents oc-
curs for systems with a fixed concentration
cif the slope of the critical line vanishes,
dT
c(c)/dc= 0, for the considered concen-
tration. This is approximately true for b-
brass, for instance.
4.2.3 Second-Order Versus First-Order
Transitions; Tricritical and Other
Multicritical Phenomena
An important question for any phase
transition is deciding a prioriwhether it
should be a second- or first-order transi-
tion; general principles are sought in an-
swering this question, such that there
would be no need for specific experimental
data.
In the Landau theory, general symme-
try conditions exist which allow second-www.iran-mavad.com
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322. order transitions (Lifshitz, 1942). We here
merely state them, without even attempting
to explain the group-theoretical language
(see, e.g., Tolédano and Tolédano, 1987,
for a thorough treatment):
(i) The order parameters 
Ftransform as
a basis of a single irreducible repre-
sentation Xof the group G
0character-
izing the symmetry properties of the
disordered phase.
(ii) The symmetric part of the representa-
tion X
3
should not contain the unit
representation.
(iii) If the antisymmetric part of X
2
has a
representation, the wavevector q asso-
ciated with X is notdetermined by
symmetry. In this case q is expected 
to vary continuously in the ordered
phase.
If these conditions are met, a transition
can nevertheless be first order, because a
fourth-order term can be negative (see Sec.
4.2.1). If they are not met, the transition
must be first order, according to the Lan-
dau rules.
The first of these rules essentially says
that if in the ordered phase two quantities
essentially independent of each other (not
related by any symmetry operation, etc.)
play the role of order parameter compo-
nents 
F
1, F
2, there is no reason why in the
quadratic term r
1(T)F
1
2+r
2(T)F
2
2of the
Landau expansion the coefficients r
1(T),
r
2(T) should change their sign at the same
temperature. Thus, if 
F
1, F
2are “primary”
order parameter components, they should
appear via a first-order transition.
The second condition essentially implies
the absence of third-order terms (
F
3
) in the
Landau expansion. It turns out, however,
that in d = 2 dimensions there are well-
known counter-examples to these rules,
namely the Potts model (Potts, 1952) with
q= 3 and q= 4 (see Eq. (4-20)); as shown
by Baxter (1973), these models have sec-
ond-order transitions. The critical expo-
nents for these models are now believed 
to be known exactly, e.g., (q=3)
a=1/3,
b=1/9, g=13/9 and (q=4) a=2/3,
b=1/12, g= 7/6 (Den Nijs, 1979; Nienhuis
et al., 1980). As mentioned in Sec. 4.2.1, an
experimental example for the three-state
Potts model is the ÷
–
3 superstructure of var-
ious adsorbates on graphite. Convincing
experimental evidence for a specific heat
divergence of He
4
on grafoil described by
a=1/3 was presented by Bretz (1977). The
system O on Ru (001) in the p (2¥2) struc-
ture (Piercy and Pfnür, 1987) falls into the
class of the four-state Potts models and
again the data are in reasonable agreement
with the predicted critical exponents.
In d= 3 dimensions, however, the three-
states Potts model has a weak first-order
transition (Blöte and Swendsen, 1979) so
that the Landau rule (ii) is not violated.
This is of relevance for metallurgical sys-
tems such as the CuAu ordering (Fig. 4-8)
on the f.c.c. lattice, which, according to
Domany et al. (1982), belongs to the class
of the three-states Potts model, while the
Cu
3Au structure belongs to the class of the
four-state Potts model. Another example of
the three-state Potts model is the structural
transition of SrTiO
3stressed in the [111]
direction (Bruce and Aharony, 1975). The
order–disorder transition for all systems
described by these types of ordering are al-
ways of first order, so there is no contradic-
tion with this Landau rule, except for the
transition to a charge density wave state in
2H-TaSe
2(Moncton et al., 1977) which ap-
parently is second order although there is a
third-order invariant (Bak and Mukamel,
1979). It is possible, of course, that the
transition is very weakly first order so that
experimentally it could not be distin-
guished from second order. This remark
also holds for other examples of apparent
270 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

323. 4.2 Phenomenological Concepts 271
violations of the Landau rules discussed by
Tolédano and Pascoli (1980), Tolédano
(1981), and Tolédano and Tolédano (1987).
There are many transitions which the
above Landau symmetry criteria would
permit to be of second order but which are
actually observed to be first order. Exam-
ples are type I or type II antiferromagnetic
structures, consisting of ferromagnetic
(100) or (111) sheets, respectively, with an
alternating magnetic moment direction be-
tween adjacent sheets (cf. Figs. 4-4 and 
4-8a). Experimentally first-order transi-
tions are known for FeO (Roth, 1958), TbP
(Kötzler et al., 1979) (these systems have
order parameter dimensionality n=4), UO
2
(Frazer et al., 1965; this is an example with
n= 6), MnO (Bloch and Mauri, 1973), NiO
(Kleemann et al., 1980) (examples with
n= 8), etc. The standard phenomenological
understanding of first-order transitions in
these materials invoked magnetostrictive
couplings (Bean and Rodbell, 1962) or
crystal field effects (in the case of UO
2; see
Blume, 1966), which make the coefficient
uin Eq. (4-10) negative and thus produce a
free energy of the type shown in Fig. 4-6b.
It has been suggested that the first-order
character of the phase transition in these
materials is a fundamental property due 
to the large number n of order-parameter
components and the symmetry of the 
Hamiltonian: renormalization group ex-
pansions discussed by Mukamel et al.
(1976a, b) and others suggested that all
such transitions are “fluctuation-induced
first-order transitions”, i.e., all antiferro-
magnets with n≥4 order parameters must
have first-order transitions. However, there
seem to be many counter-examples of cu-
bic n≥4 antiferromagnets with phase tran-
sitions of apparently second order, such as
BiTb (Kötzler, 1984), GdTb (n= 4; Hulli-
ger and Siegrist, 1979), and GdSb (n=8;
McGuire et al., 1969). The reasons for 
this discrepancy between renormalization
group predictions and experiments are not
clear.
There are various other cases where
“fluctuation-induced first-order transitions”
occur, as reviewed in detail by Binder
(1987a). Here we mention only the exam-
ple of the phase transition from disordered
block copolymer melts to the lamellar mes-
ophase (Fig. 4-7b). For this system the
Figure 4-8.Five crystallographic superstructures on
the face-centered cubic (fcc) lattice and their equiva-
lent antiferromagnetic superstructures if in a binary
alloy A-atoms (full circles) are represented as “spin
up” and B atoms (open circles) are represented as 
“spin down”. Case (a) refers to the CuAuI structure,
(b) to the A
2B
2structure, (c) to the A
3B structure of
Cu
3Au type, (d) to the A
3B structure of Al
3Ti type,
and (e) the A
2B structure as it occurs in Ni
2V or
Ni
2Cr. In case (a) the labels 1, 2, 3, and 4 indicate the
decomposition of the fcc lattice into four interpene-
trating simple cubic (sc) sublattices.www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

324. free-energy functional does not attain its
minimum at wavevector q= 0 in reciprocal
space but on a surface given by the equa-
tion |q|=q*=2p /
l, lbeing the wave-
length of the lamellar pattern. This reflects
the degeneracy that there is no preferred di-
rection and hence the lamellae may be
oriented in any direction. Owing to the
large “phase space” in q-space where the
inverse of the wavevector-dependent sus-
ceptibility goes “soft” (
c
–1
(|q|=q*)Æ0
as TÆT
c), the mean-square amplitude of
the local fluctuation of the order parameter
·
F
2
Ò–·FÒ
2
would diverge as TÆT
c,
which is physically impossible. It can be
shown that this difficulty is avoided be-
cause the strong local order parameter fluc-
tuations turn unegative near T
c(Brazov-
skii, 1975; Fredrickson and Helfand, 1987;
Fredrickson and Binder, 1989), producing
a free energy, as shown in Fig. 4-6b, for the
case of “symmetric” diblock copolymers
where mean field theory would predict a
second-order transition. Experiments seem
to confirm the first-order character of the
transition (Bates et al., 1988).
For certain systems, for which u>0 and
thus a second-order transition occurs, it is
possible by variation of a non-ordering
field to change the sign of the coefficient u
at a particular point. This temperature T
t,
where uvanishes in Eq. (4-14), is called a
tricritical point. From Eq. (4-14) we then
immediately find
(4-86)
F
0= (–r/v)
1/4
= (r¢/v)
1/4
(T
t/T–1)
1/4
Hence the tricritical order parameter expo-
nent 
b
t=1/4, while Eqs. (4-32) to (4-48) re-
main unchanged and thus 
g
t=1, n
t=1/2,
h
t= 0. The critical isotherm, however, be-
comes v
F
0
5=H/k
BT, i.e., d
t= 5. From Eq.
(4-15) we finally find
(4-87)
FF
Vk T
tt() ()
/
F
0
12 0
3
−
−
−⎛
⎝
⎞
⎠
B
=
v
which implies a specific heat divergence
C~(–t)
–1/2
, i.e., a
t=1/2. We note that this
set of exponents also satisfies the scaling
laws in Eqs. (4-78), (4-81), (4-83), and 
(4-84). Moreover, using this set of expo-
nents in the Ginzburg criterion, Eqs. (4-58)
to (4-61), we find that, with the Landau ex-
ponents for a tricritical point, this condi-
tion is marginally fulfilled, unlike the case
of ordinary critical points (Eq. (4-62)). It
turns out that d* = 3 is the marginal dimen-
sion for tricritical points, and the mean-
field power laws are modified by logarith-
mic correction terms similar to those noted
for dipolar systems (Eq. (4-72)). Experi-
mental examples for tricritical points in-
clude strongly anisotropic antiferromagnets
in a uniform magnetic field (e.g., FeCl
2,
(Dillon et al., 1978), and dysprosium alu-
minum garnet (DAG) (Giordano and Wolf,
1975), see Fig. 4-9 for schematic phase 
diagrams), systems undergoing structural
phasetransitions under suitable applied pres-
sure, such as NH
4Cl which has a tricritical
point at T
t= 250 K and p
t= 128 bar (Yelon et
al., 1974) or the ferroelectric KDP at T
t=113
K and p
t= 2.4 kbar (Schmidt, 1978), etc. A
model system which has been particularly
carefully studied is He
3
–He
4
mixtures: the
transition temperature T
l(x) of the normal
fluid–superfluid He
4
is depressed with in-
creasing relative concentration xuntil, at a
tricritical point T
t=T
l(x
t), the transition be-
comes first order. This then implies a phase
separation between a superfluid phase with
a lower He
3
content and a normal fluid He
3
-
rich phase. Most common, of course, are
tricritical phenomena in fluid binary mix-
tures arising from the competition of gas–
liquid transitions and fluid–fluid phase
separations in these systems (Scott, 1987).
Just as the vanishing coefficient uin Eq.
(4-10) leads to a multicritical point, the tri-
critical point, another multicritical point is
associated with the vanishing of the coeffi-
272 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

325. 4.2 Phenomenological Concepts 273
cient of the [— F(x)]
2
term; since this term
was also introduced as the lowest-order
term of a systematic expansion, higher-or-
der terms such as [—
2
F(x)]
2
must then be
included in Eq. (4-10). This problem is
most conveniently discussed in the frame-
work of the wavevector-dependent suscep-
tibility 
c(q), Eq. (4-34), which including
the contribution resulting from a term 
[—
2
F(x)]
2
can be written as (t
0=1–T
L/T)
(4-88)
where K
1and K
2are phenomenological co-
efficients. If K
1< 0, the divergence of c(q)
does not occur for t
0= 0, q= 0 but at a value
q
maxfound from:
(4-89)
d
d
=
=
q
tKqKq
qKK
()
(/)
max
/
01
2
2
4
12
12
0
2
++
⇒−
c()
ˆ
q=
G
0
01
2
2
4
tKqKq++
The peak height of 
c(q) for | q|=q
maxis
then described by
(4-90)
which implies that the actual critical point occurs at T
c=T
L[1+K
1
2/(4K
2)]. An exam-
ple of such an ordering characterized by a nonzero wavenumber q*=q
maxis the meso-
phase formation in block copolymer melts (Fig. 4-7) (see Leibler (1980)). Related phe- nomena also occur in crystals exhibiting “incommensurate superstructures”. Usu-
ally
c(q) is then not isotropic in q-space; in
uniaxial systems, Eqs. (4-88) to (4-90) then apply only if qis parallel to this preferred
direction. The superstructure described by the wavelength 
l=2p/q
maxis determined
by the coefficients K
1and K
2and thus in
general it is not a simple multiple of the lat- tice spacing a , but (
l/a) is an irrational
number. This is what is meant by the term
incommensurate. Examples of such incom- mensurate structures are “helimagnetic structures” such as VF
2, MnAu
2, Eu, Ho
and Dy (Tolédano and Tolédano, 1987). Note that in these systems the reason for the incommensurate structure is not the  existence of a negative coefficient in front of the [—
F(x)]
2
term but the existence of 
terms such as  where
F
+
=F
0cos (p z/c),  F
–
=F
0sin (pz/c), c
being the lattice spacing in the z-direction.
Such terms involving linear terms in —
F(x)
are allowed by symmetry in certain space groups and are called “Lifshitz invariants”. Moving from lattice plane to lattice plane in the z-direction in such a helimagnetic
spin structure, the spin vector (pointing perpendicular to z) describes a spiral struc-
ture like a spiral staircase. Of course, such incommensurate superstructures (also called “modulated phases”) are not restrict-
FF
FF
FF
FF
+
−
−
+∂
∂
−
∂
∂⎛
⎝
⎜
⎞
⎠
⎟
zz
,
c()
ˆ
/( )
maxq
tK K
=
G
01
2
2 4−
Figure 4-9.Schematic phase diagrams of antiferro-
magnets with uniaxial anisotropy in an applied uni-
form magnetic field H
||applied parallel to the easy
axis: Case (a) shows the case of weak anisotropy,
case (b) the case of intermediate anisotropy, where in
addition to the antiferromagnetic ordering of the spin
components in the direction of the easy axis, a spin-
flop ordering of the transverse components also oc-
curs. In case (a) both transitions T
||(H
||), and T
^(H
||)
are of second order and meet in a bicritical point. For
intermediate strength of the anisotropy the line
T
^(H
||) does not end at the bicritical point, but rather
in a critical end-point at the first-order transition line.
Then a tricritical point also appears where the anti-
ferromagnetic transition T
||(H
||) becomes first order.
For very strong anisotropy, the spin-flop phase disap-
pears altogether.www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

326. ed to magnetic systems, and many exam-
ples of dielectric incommensurate phases
have been identified (e.g., NaNO
2(Tani-
saki, 1961), Rb
2ZnCl
4, BaMnF
4, thiourea
[SC(NH
2)
2], see Tolédano and Tolédano
(1987)). Whereas Rb
2ZnCl
4and BaMnF
4
also involve the existence of Lifshitz invar-
iants, thiourea is an example of the case
discussed in Eqs. (4-88) to (4-90). Various
examples of modulated superstructures ex-
ist in metallic alloys; see Selke (1988,
1989, 1992) and De Fontaine and Kulik
(1985) for reviews of the pertinent theory
and experimental examples such as Al
3Ti
(Loiseau et al., 1985) and Cu
3Pd (Broddin
et al., 1986).
Whereas for TeT
cthe ordering of the
incommensurate phase can be described in
terms of a sinusoidal variation of the local
order parameter density 
F(x), the nonlin-
ear terms present in Eq. (4-10) at lower
temperatures imply that higher-order har-
monics become increasingly important.
Rather than a sinusoidal variation, the
structure is then better described in terms
of a periodic pattern of domain walls (or
“solitons”), and we talk about a “multi-
soliton lattice”or “soliton staircase”. An-
other important fact about modulated
phases is that the wavevector characteriz-
ing the modulation period is not fixed at
q
max, but varies with temperature or other
parameters of the problem. This is ex-
pected from the third of the general Landau
rules formulated at the beginning of Sec.
4.2.3. Usually this variation of qstops at
some commensurate value where a “lock-
in transition” of the incommensurate struc-
tures occurs. In the incommensurate re-
gime rational values of the modulation 
(“long-period superstructures”) may have
an extended regime of stability, leading to
an irregular staircase-like behavior of the
modulation wavelength 
las a function of
temperature (or other control parameters
such as pressure, magnetic field (for mag-
netic structures), concentration (for alloys;
see Fig. 4-10), etc.). Theoretically under
certain conditions even a staircase with an
infinite number (mostly extremely small!)
of steps can be expected (“devil’s stair-
case”, see Selke (1988, 1989, 1992) for a
discussion and further references). We also
emphasize that the modulation need not in-
volve only one direction in space (one
wavevector), but can involve several wave-
274 4 Statistical Theories of Phase Transitions
Figure 4-10.Period M(in units of the lattice spacing
in the modulation direction) for the alloy Al
3–xTi
1+x
as a function of the annealing temperature, as ob-
tained from visual inspection of high-resolution elec-
tron microscopic images. Each of the fifteen different
commensurate superlattices observed is composed of
antiphase domains of length one or two (1-bands or
2-bands, as indicated by sequences ·211Ò, ·21Òetc.)
based on the L1
2structure. Very long annealing times
were needed to produce these long-period super-
structures supposedly at equilibrium. From Loiseau
et al. (1985).www.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

327. 4.2 Phenomenological Concepts 275
vectors. An example of the case of a two-
rather than one-dimensional modulation is
LiKSO
4(Pimenta et al., 1989).
Here we discuss only the special case 
of Eqs. (4-88)–(4-90) briefly where, by
changing external control parameters, it 
is possible to reach a multicritical point
where K
1vanishes, the so-called “Lifshitz
point” (Hornreich et al., 1975). From Eq.
(4-88) we then find that for T=T
c(=T
L) we
have 
c(q)~q
–4
, and comparing this with
the result following from a Fourier trans-
formation of Eq. (4-47), 
c(q)
T
c
~q
–2+h
, 
we conclude that 
h
L= – 2 for the Lifshitz
point, whereas for T>T
cwe still have
c(q=0)=G
ˆ
0t
0
–1, i.e., g
L= 1. Since for
K
1= 0 Eq. (4-88) can be rewritten as
c(q) = G
ˆ
0(1 – T
L/T)
–1
/[1+ x
4
q
4
]
x= [K
2/(1 – T
L/T)]
1/4
(4-91)
we conclude that 
n
L= 1/4 for the correla-
tion length exponent at the Lifshitz point.
Also for the Lifshitz point we can ask
whether the Landau description presented
here is accurate, and whether or not statis-
tical fluctuations modify the picture. The
result is that the critical fluctuations are
very important. A marginal dimensionality
d* = 8 results in the case of an “isotropic
Lifshitz point”, where in
K
1x(∂F/∂x)
2
+K
1y(∂F/∂y)
2
+K
1z(∂F/∂z)
2
we have simultaneous vanishing of all co-
efficients K
1x, K
1y, and K
1z. Note that the
more common case is the “uniaxial Lifshitz
point”, where only K
1zvanishes while K
1x
and K
1yremain nonzero. In this case the
multicritical behavior is anisotropic, and
we must distinguish between the correla-
tion length 
x
||describing the correlation
function decay in the z-direction and the
correlation length 
x
^in the other direc-
tions. Eq. (4-91) then holds for 
x
||only,
whereas Eq. (4-39) still holds for 
x
^. 
While experimental examples for Lif-
shitz points are extremely scarce, e.g., the
structural transition in RbCaF
3under (100)
stress (Buzaré et al., 1979), more common
multicritical points are bicritical points
where two different second-order transi-
tion lines meet. Consider, for example, the
generalization of Eq. (4-10) to an n-compo-
nent order-parameter field, but with differ-
ent coefficient r
ifor each order-parameter
component 
F
i:
For simplicity, the fourth-order term has
been taken as fully isotropic. If r
1=r
2=…
=r
n, we would have the isotropic n-vector
model discussed in Secs. 4.2.2 and 4.2.3.
We now consider the case where some of
these coefficients differ from each other,
but a parameter pexists on which these co-
efficients depend in addition to the temper-
ature. The nature of the ordering will be 
determined by the term r
iF
i
2, for which 
the coefficient r
ichanges sign at the high-
est temperature. Suppose this is the case
where i=1 for p<p
b, then we have a one-
component Ising-type transition at T
c1(p)
given by r
1(p,T) = 0 (the other compo-
nents 
F
ibeing “secondary order parame-
ters” in this phase). If, however, for p>p
b
the coefficient r
2(p,T) = 0 at the highest
temperature T
c2(p), it is the component F
2
which drives the transition as the primary
order parameter. The point p=p
b, T
c1(p)=
T
c2(p)=T
bis then called a bicritical point.
An example of this behavior is found for
weakly uniaxial antiferromagnets in a uni-
1
2
1
2
1
2
1
24
2
0
11
2
22
2
222
2
1
22
kT
F
kT
rr
r
u
R
d
nn
n
BB
=( 4-9)
d⎧[()]
() ()
() [ ()]
[( ) ( ) ]FF
FFx
xx x
xx
++
⎧
⎨
⎩
+…
++
+∇+…+∇
⎫
⎬
⎭
∫ FF
F
FFwww.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

328. form magnetic field H
||(Fig. 4-9). A model
Hamiltonian for such a system can be writ-
ten as
where we assume J< 0 (antiferromagne-
tism) and for 
D> 0 the easy axis is the z-
axis. For small H
||, we have a uniaxial anti-
ferromagnetic structure and the order pa-
rameter is the z-component of the stag-
gered magnetization M
s
z. For stronger
fields H
||, however, we have a transition 
to a spin-flop structure (two-component 
∫=
(4-93)
−− +
+−
〈〉
∑
∑
ij
i
x
j
x
i
y
j
y
i
z
j
z
i
i
z
JSSSS
JS S H S
,
{( )[ ]
}
1D
||
order parameter, due to the perpendicu- lar components M
s
xand M
s
yof the stag-
gered magnetization). In the framework of Eq. (4-92), this would mean n=3 and
r
2∫r
3≠r
1, with H
||being the parameter p.
Both lines T
N
||(H
||) and T
N
^(H
||) and the
first-order line between the two antiferro- magnetic structures join in a bicritical point. A well known example of such a be- havior is GdAlO
3with T
b= 3.1242 K (Roh-
rer, 1975). Another example, in our opin- ion, is the Fe–Al system where the ferro- paramagnetic critical line joins the second- order A2–B2 transition (Fig. 4-11). In the metallurgical literature, the magnetic tran-
276 4 Statistical Theories of Phase Transitions
Figure 4-11.The Fe–Al phase diagram, as obtained from a mean-field calculation by Semenovskaya (1974) 
(left), and experimentally by Swann et al. (1972) (right). The ferro-paramagnetic transition is shown by the
dash-dotted line. The crystallographically disordered (A2) phase is denoted as 
a
nin the nonmagnetic and a
min
the ferromagnetic state. The ordered FeAl phase having the B2 structure (Fig. 4-3) is denoted as 
a
2, and the or-
dered Fe
3Al phase having the D0
3structure (Fig. 4-3) is denoted as a
1nor a
1m, depending on whether it is non-
magnetic or ferromagnetically ordered, respectively. Note that first-order transitions in this phase diagram are
associated with two-phase regions, since the abscissa variable is the density of an extensive variable, unlike the
ordinate variable in Fig. 4-9. From Semenovskaya (1974).www.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

329. 4.2 Phenomenological Concepts 277
sition of Fe–Al alloys is often disregarded
and then the multicritical point where the
A2–B2 transitions change from second to
first order is interpreted as a tricritical
point. Bicritical phenomena for structural
phase transitions are also known, such as
SrTiO
3under stress in the [001] direction 
(Müller and Berlinger, 1975).
The structure of the phase diagram near
bicritical points has been analyzed in detail
by renormalization group methods (Fisher
and Nelson, 1974; Fisher, 1975a, b; Muka-
mel et al., 1976a, b), experiment (Rohrer
and Gerber, 1977) and Monte Carlo simu-
lations (Landau and Binder, 1978). Apart
from the fact that the exponents introduced
so far take different values as TÆT
b, there
is one additional exponent, the “crossover
exponent” 
F, describing the singular ap-
proach of phase-transition lines towards
the multicritical point. This behavior, and
also other more complicated phenomena
(tetracritical points, tricritical Lifshitz
points, etc.) are beyond the scope of this
chapter. For an excellent introduction to
the field of multicritical phenomena we re-
fer to Gebhardt and Krey (1979) and for a
comprehensive review to Pynn and Skjel-
torp (1983).
4.2.4 Dynamics of Fluctuations
at Phase Transitions
Second-order phase transitions also
show up in the “critical slowing down”of
the critical fluctuations. In structural phase
transitions, we speak about “soft phonon
modes”; in isotropic magnets magnon
modes soften as Tapproaches T
cfrom be-
low; near the critical point of mixtures the
interdiffusion is slowed down; etc. This
critical behavior of the dynamics of fluctu-
ations is again characterized by a dynamic
critical exponent z; we expect there to be
some characteristic time 
twhich diverges
as TÆT
c:
t~ x
z
~ |1– T/T
c|
–nz
(4-94)
Many of the concepts developed for static
critical phenomena (scaling laws, univer-
sality, etc.) can be carried over to dynamic
critical phenomena: “dynamic scaling”
(Halperin and Hohenberg, 1967; Ferrell et
al., 1967; Hohenberg and Halperin, 1977),
“dynamic universality classes” (Halperin
et al., 1974, 1976; Hohenberg and Halpe-
rin, 1977), etc. However, the situation is
much more complicated because the static
universality classes are split up in the dy-
namic case, depending on the nature of the
conservation laws, mode coupling terms in
the basic dynamic equations, etc. For ex-
ample, we have emphasized that aniso-
tropic magnets such as RbMnF
3, ordering
alloys such as CuZn, unmixing solid mix-
tures such as Al– Zn, unmixing fluid mix-
tures such as lutidine–water, and the gas–
fluid critical point all belong to the same
Ising-like static universality class, but each
of these examples belongs to a different dy-
namic universality class! Thus, in the an-
isotropic antiferromagnet, no conservation
law needs to be considered, whereas the
conservation of concentration matters for
all mixtures (where it means the order pa-
rameter is conserved) and for ordering al-
loys (where the order parameter is not con-
served but coupled to a conserved “nonor-
dering density”). Whereas in solid mix-
tures the local concentration relaxes simply
by diffusion, in fluid mixtures hydrody-
namic flow effects matter, and also play a
role at the liquid–gas critical point. In the
latter instance, conservation of energy
needs to be considered; it does not play a
role for phase transitions in solid mixtures,
of course, where the phonons act as a “heat
bath” to the considered configurational de-
grees of freedom.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

330. We do not give a detailed account of crit-
ical dynamics here, but discuss only the
simplest phenomenological approach (van
Hove, 1954) for non-conserved order pa-
rameters. We consider a deviation of the or-
der parameter D
F(x,t) from equilibrium at
space xand time t, and ask how this devia-
tion decays back to equilibrium. The stan-
dard assumption of irreversible thermody-
namics is a relaxation assumption; we put
(4-95)
where 
G
0is a phenomenological rate factor.
Using Eq. (4-10) and linearizing 
F(x,t)=
F
0+DF(x,t) around F
0yields
Assuming that D
F(x,t) is produced by 
a field H (x)=H
qexp (iq∙x) which is
switched off at time t= 0, we obtain (see
also Sec. 4.2.2)
(4-97)
where Eq. (4-34) has been used. Since 
c(q)
diverges for q= 0, TÆT
c, the characteris-
tic frequency 
w(q=0)∫ t
–1
vanishes as
w(q=0)~c
T
–1~x
–g/n
=x
–(2–h)
, and by
comparison with Eq. (4-94) the result of
the “conventional theory” is obtained:
z= 2 – 
h (4-98)
Although Eq. (4-98) suggests that there is a
scaling relationship linking the dynamic to
the static exponents, this is not true in gen-
eral if effects due to critical fluctuations are
taken into account. In fact, for noncon-
served systems, zslightly exceeds 2 –
hand
extensive Monte Carlo studies were needed
D
DF
F
G
q
q
qq q
()
()
exp [ ( ) ], ( ) / ( )
t
t
0
0==− ww c
∂
∂
−
{+
−∇
⎫
⎬
⎭
t
tru t
R
d
t
DD
DFGFF
F(,) ( ) (,)
[(,)]
xx
x
=
(4-96)
00
2
2
2 3
∂
∂
−
∂
∂t
tD
D
FG
F(,)
()
x
x
=
0
⎧
to establish its precise value (Wansleben
and Landau, 1987).
It is important to realize that not all fluc-
tuations slow downas a critical point is ap-
proached, but only those associated with
long-wavelength order parameter varia-
tions. This clearly expressed in terms of the
dynamic scaling relationship of the charac-
teristic frequency:
w(q) = q
z
w˜(′),′= q x (4-99)
where 
w˜(′) is a scaling function with the
properties 
w˜(•) = const. and w˜(′O1) ~′
–z
,
and, hence, is consistent with Eq. (4-94).
Most experimental evidence on dynamic
critical phenomena comes from methods
such as inelastic scattering of neutrons or
light, NMR or ESR spectroscopy, and
ultrasonic attenuation. Although the over-
all agreement between theory and experi-
ment is satisfactory, only a small fraction
of the various theoretical predictions have
been thoroughly tested so far. For reviews,
see Hohenberg and Halperin (1977) and
Enz (1979).
Finally, we also draw attention to nonlin-
ear critical relaxation. Consider, for exam-
ple, an ordering alloy which is held well
below T
cin the ordered phase and assume
that it is suddenly heated to T=T
c. The or-
der parameter is then expected to decay to-
wards zero with time in a singular fashion,
F(t)~t
–b/nz
(Fisher and Racz, 1976). Sim-
ilarly, if the alloy is quenched from a tem-
perature T>T
cto T=T
c, the superstructure
peak appears and grows in a singular fash-
ion with the time t after the quench,
S(q,
t)~t
g/zn
(Sadiq and Binder, 1984). Re-
lated predictions also exist for the nonlin-
ear relaxation of other quantities (ordering
energy, electrical resistivity) (Binder and
Stauffer, 1976a; Sadiq and Binder, 1984).
A very important topic is the dynamics
of first-order phase transitions, which will
not be discussed here because it is dis-
278 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

331. 4.2 Phenomenological Concepts 279
cussed elsewhere in this book (see Chap-
ters by Binder and Fratzl (2001) and by
Wagner et al. (2001)).
4.2.5 Effects of Surfaces and of Quenched
Disorder on Phase Transitions:
A Brief Overview
It a system contains some impurity at-
oms which are mobile, we refer to “an-
nealed disorder” (Brout, 1959). From the
point of view of statistical thermodynam-
ics, such mobile impurities act like addi-
tional components constituting the system
under consideration. For a second-order
transition, small fractions of such addi-
tional components have rather minor ef-
fects: the transition point may be slightly
shifted relative to that of the pure material,
and as the concentration of these impurities
is normally strictly conserved, in principle
the so-called “Fisher renormalization” 
(Fisher, 1968) of critical exponents is ex-
pected, as discussed in Sec. 4.2.2. For
small impurity contents, the region around
the transition point where this happens is
extremely narrow, and hence this effect is
not important.
The effect of such impurities on first-or-
der transitions is usually more important,
e.g., for a pure one-component system, the
melting transition occurs at one well-de-
fined melting temperature, whereas in the
presence of mobile impurities this transi-
tion splits up into two points, correspond-
ing to the “solidus line” and “liquidus line”
in the phase diagram of a two-component
system. This splitting for small impurity
contents is simply proportional to the con-
centration of the impurities of a given type.
In practice, where a material may contain
various impurities of different chemical na-
tures and different concentrations which
often are not known, these splittings of the
transition line in a high-dimensional pa-
rameter space of the corresponding multi-
component system show up like a rounding
of the first-order transition.
In solids, however, diffusion of impurity
atoms is often negligibly small and such a
disorder due to frozen-in, immobile impur-
ities is called “quenched disorder” (Brout,
1959). Other examples of quenched disor-
der in solids are due to vacancies, in addi-
tion to extended defects such as disloca-
tions, grain boundaries, and external sur-
faces. Quenched disorder has a drastic effect
on phase transitions, as will be seen below.
Since the disorder in a system is usually
assumed to be random (irregular arrange-
ment of frozen-in impurities, for example),
we wish to take an average of this random
disorder. We denote this averaging over the
disorder variables, which we formally de-
note here as the set {x
a}, by the symbol
[…]
avin order to distinguish it from the
thermal averaging ·…Ò
Timplied by statisti-
cal mechanics. Thus the average free en-
ergy which needs to be calculated in the
presence of quenched disorder is 
whereas in the case of annealed disorder it
would be the partition function rather than
the free energy which is averaged over the
disorder:
Eq. (4-101) again shows that annealed dis-
order just means that we enlarge the space
of microscopic variables which need to be
included in the statistical mechanics treat-
ment, whereas the nature of quenched aver-
aging (Eq. (4-100)) is different.
FkTZx
kT x kT
kT x kT
i
i
i
x
i
ann B av
BBav
BB
=( 4-1)
=tr
=t r
−
−
(
− )
− (
− )
ln[ { }]
ln [exp( { , }/ )]
ln exp( { , }/ )
{}
{, }
a
a
a
a
01
F
F
F
F
∫
∫
FkTZx
kT x kT
i
i
=( 4-1)
=tr
Ba v
BB
av−
−
[(
− )]
[ln { }]
ln exp( { , }/ )
{}
a
a 00
F
F∫www.iran-mavad.com
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332. Experiments can be carried out with just
one probe crystal and no averaging over
many samples as is implied by Eq. (4-100)
needs to be performed. This is because we
can think about the parts of a macroscopic
sample as subsystems over which such an
average as written in Eq. (4-100) is actu-
ally performed. This “self-averaging”
property of macroscopically large systems
means that the average […]
avover the set
{x
a} could also be omitted: it need only be
kept for mathematical convenience.
We first discuss the physical effects of
randomly quenched impurities (point-like
defects!) on phase transitions, assuming
the concentrations of these defects to be
very small. Various cases need to be distin-
guished, depending on the nature of the 
local coupling between the defect and the
local order parameter. This coupling may
be linear (“random field”), bilinear (“ran-
dom bond”) or quadratic (“random aniso-
tropy”). As an example, let us consider
magnetic phase transitions described by the
n-vector model of magnetism, with an ex-
change interaction depending on the dis-
tance between lattice points a
iand a
j:
(4-102)
In Eq. (4-102) we have assumed that im-
purities produce a random field H
iwhich
has the properties
[H
i]
av= 0 , [H
i
2]
av= H
2
R
(4-103)
and S
iis an n-component unit vector in the
direction of the local order parameter. Now
we can show that for n≥2 arbitrary small
amplitudes of the random field H
R(which
physically is equivalent to arbitrary small
impurity concentration) can destroy uni-
form long-range order (Imry and Ma,
1975): the system is broken into an irregu-
lar arrangement of domains. The mean 
size of the domains is larger as H
R(or the
∫=−−⋅−
≠
∑∑
ij
ijij
i
ii
z
JH S()aaSS
impurity concentration, respectively) is smaller. Since no ideal long-range order is established at the critical point, this corre- sponds to a rounding of the phase transi- tion.
In the case n=1 and d= 3 dimensions,
very weak random fields do not destroy uniform long-range order (Imbrie, 1984; Nattermann, 1998), although long-range order is destroyed if H
Rexceeds a certain
threshold value H
c
R
(T) which vanishes as
TÆT
c. But the random field drastically
changes the nature of both static and dy- namic phenomena (Villain, 1985; Grin- stein, 1985). In d= 2 dimensions, arbitrary
weak random fields destroy even Ising- like (n=1) order, producing domains of
size 
x
Dwith lnx
D~(J/H
R)
2
(see Binder,
1983a).
We now turn briefly to the physical real-
ization of “random fields”. For symmetry reasons, no such defects are expected for ferromagnets, but they may be induced in- directly by a uniform field acting on an antiferromagnet (Fishman and Aharony, 1979; Belanger, 1998). Since antiferro- magnets in a field can be “translated” into models for order–disorder phenomena in alloys (the magnetization of the antiferro- magnet “translates” into the concentration of the alloy, see Sec. 4.3.1) or (in d=2 di-
mensions) in adsorbed layers (the magnet- ization then “translates” into the coverage of the layer), random-field effects are im- portant for ordering alloys or order–disor- der transitions in monolayers, also. A nice experimental example was provided by Wiechert and Arlt (1993) who showed that the transition of CO monolayers on graph- ite near 5 K to a ferrielectric phase is rounded by small dilution with N
2mole-
cules, in accord with the theory of random- field effects. Furthermore, random fields act on many structural phase transitions owing to impurity effects: ions at low-sym-
280 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

333. 4.2 Phenomenological Concepts 281
metry, off-center positions may produce
both a random electric field and a random
strain field. Also, symmetry generally per-
mits random fields to act on tensorial order 
parameters in diluted molecular crystals
(Harris and Meyer, 1985). For diluted mo-
lecular crystals (such as N
2diluted with Ar,
or KCN diluted with KBr, etc.) it is still un-
clear whether the resulting “orientational
glass” phase is due to these random fields
or to random bonds (for reviews, see Loidl
(1989); Höchli et al. (1990); Binder and
Reger (1992); Binder (1998)).
The “random bond” Hamiltonian differs
from Eq. (4-102) by the introduction of dis-
order into the exchange terms rather than
the “Zeeman energy”-type term,
(4-104)
Whereas in Eq. (4-102) we have assumed
the pairwise energy to be translationally in-
variant (the exchange energy Jdepends
only on the distance a
i–a
jbetween the
spins but not on their lattice vectors a
iand
a
jseparately), we now assume J
ijto be a
random variable, e.g., distributed accord-
ing to a Gaussian distribution
P(J
ij) µexp [– (J
ij– J
–
)
2
/2(DJ)
2
] (4-105)
or according to a two-delta function distri-
bution
P(J
ij) =pd(J
ij– J) + (1 – p) d(J
ij) (4-106)
If J
–
oDJ(Eq. (4-105)) or if 1–pO1 (Eq.
(4-106)), the ferromagnetic order occurring
for Eq. (4-104) if J> 0 or J
–
> 0 is only
weakly disturbed, both for d= 3 and for
d= 2 dimensions. Following an argument
presented by Harris (1974), we can see that
the critical behavior of the system remains
unaltered in the presence of such impurities
provided that the specific heat exponent
a< 0, whereas a new type of critical behav-
ior occurs for 
a≥0 (see Grinstein (1985)
∫=−⋅−
≠
∑∑
ij
ij i j
i
i
z
JHSSS
for a review). Of course, the impurities will always produce some shift of the critical temperature, which decreases as pin Eq.
(4-106) decreases. When pbecomes
smaller than a critical threshold value p
c,
only finite clusters of spins are still mutu- ally connected by nonzero exchange bonds J> 0, and long-range order is no longer
possible. The phase transition at T= 0 pro-
duced by variation of p in Eq. (4-106) can
hence be interpreted purely geometrically in terms of the connectivity of finite clus- ters or an infinite “percolating” network of spins (Stauffer and Aharony, 1992). This transition is again described by a com- pletely different set of exponents.
For the case where |J
–
|ODJin Eq. 
(4-105), a new type of ordering occurs, which is not possible in systems without quenched disorder; a transition occurs to a state without conventional ferro- or antifer- romagnetic long-range order but to a “spin glass phase” where the spins are frozen-in in a random direction (Binder and Young, 1986; Young, 1998). The order parameter of the Ising spin glass was introduced by Edwards and Anderson (1975) as
q
EA= [·S
iÒ
2
T
]
av (4-107)
The nature of the phase transition in spin
glasses and the properties of the ordered
phase have been the subject of intense re-
search (Binder and Young, 1986). This
great interest in spin glasses can be under-
stood because many different systems show
spin glass behavior: transition metals with
a small content xof magnetic impurities
such as Au
1–xFe
xand Cu
1–xMn
x, diluted
insulating magnets such as Eu
xSr
1–xS, var-
ious amorphous alloys, and also mixed 
dielectric materials such as mixtures of
RbH
2PO
4and NH
4H
2PO
4, where the spin
represents an electric rather than magnetic
dipole moment. A related random ordering
of quadrupole moments rather than dipolewww.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

334. moments if found in “quadrupolar glasses”
(also called “orientational glasses”) such 
as K(CN)
1–xBr
xand (N
2)
xAr
1–x, (Loidl,
1989; Höchli et al., 1990; Binder and
Reger, 1992; Binder, 1998). However, it 
is still debated (Franz and Parisi, 1998)
whether or not concepts appropriate for the
description of spin glasses are also useful
for the structural glass transition of under-
cooled liquids (Jäckle, 1986).
Finally we mention systems with ran-
dom anisotropic axes which can be mod-
eled by the following Hamiltonian (Harris
et al., 1973):
(4-108)
where the e
iare vectors whose components
are independent random variables with a
Gaussian distribution. This model is also
believed to exhibit destruction of long-
range order due to break-up in domains
similar to the random field systems. It is
also suggested that spin glass phases may
occur in these systems. Again Eq. (4-108)
is expected to be relevant, not only for dis-
ordered magnetic materials but also for
dielectrics where the spin represents elec-
tric dipole moments, or displacement vec-
tors of atoms at structural phase transitions,
etc.
Very small fractions of quenched impur-
ities which do not yet have an appreciable
effect on the static critical properties of a
second-order phase transition can already
affect the critical dynamics drastically. An
example of this behavior is the occurrence
of impurity-induced “central peaks” for
structural phase transitions in the scattering
function S(q,
w) at frequency w= 0 in addi-
tion to the (damped) soft phonon peaks
(Halperin and Varma, 1976). In the above
systems where the impurities disrupt the
conventional ordering more drastically 
(such as random field systems and spin
∫=−−⋅−⋅
≠
∑∑
ij
ijij
i
ij
J() ()aaSS eS
2
glasses) the critical dynamics again have a very different character, and dynamics characteristic of thermally activated pro- cesses with a broad spectrum of relaxation times are often observed.
It is also interesting to discuss the effect
of quenched impurities on first-order tran- sitions (Imry and Wortis, 1979). It is found that typically “precursor phenomena” near the first-order transition are induced, and the latent heat associated with the transi- tion in the pure systems can be signifi- cantly reduced. It is also possible that such impurity effects may completely remove the latent heat discontinuity and lead to a rounding of the phase transition.
Such rounding effects on phase transi-
tions also occur when extended defects such as grain boundaries and surfaces are considered. These disrupt long-range order because the system is then approximately homogeneous and ideal (i.e., defect-free) only over a finite region in space. While the description of the rounding and shifting of phase transitions due to finite size ef- fects has been elaborated theoretically in detail (Privman, 1990), only a few cases exist where the theory has been tested ex- perimentally, such as the normal fluid– superfluid transitions of He
4
confined to
pores, or the melting transition of oxygen monolayers adsorbed on grafoil where the substrate is homogeneous over linear di- mension Lof the order of 100 Å (see Marx,
1989). While at critical points, a rounding and shifting of the transition normally set in when the linear dimension Land the cor-
relation length 
xare comparable (Fisher,
1971; Binder, 1987b), at first-order transi- tions the temperature region DTover
which the transition is rounded and shifted is inversely proportional to the volume of the system (Imry, 1980; Challa et al., 1986).
An important effect of extended defects
such as grain boundaries and surfaces is
282 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

335. 4.3 Computational Methods Dealing with Statistical Mechanics 283
that they often induce precursor effects to
phase transitions, e.g., the “wetting transi-
tion” (Dietrich, 1988; Sullivan and Telo da
Gama, 1985) where a fluid layer condens-
ing at a surface is a precursor phenomenon
to gas–fluid condensation in the three-
dimensional bulk volume. Similarly “sur-
face melting” and “grain-boundary melt-
ing” phenomena can be interpreted as the
intrusion of a precursor fluid layer at a sur-
face (or grain boundary) of a crystal (Di
Tolla et al., 1996). Similarly, at surfaces of
ordering alloys such as Cu
3Au, the effect
of the “missing neighbors” may destabilize
the ordering to the extent that “surface-in-
duced disordering” occurs when the system
approaches the transition temperature of
the bulk (Lipowsky, 1984). Finally, we
draw attention to the fact that the local crit-
ical behavior at surfaces differs signifi-
cantly from the critical behavior in the bulk
(Binder, 1983b). Such effects are outside
the scope of this chapter.
4.3 Computational Methods
Dealing with the Statistical
Mechanics of Phase Transitions
and Phase Diagrams
While the phenomenological theories in
Sec. 4.2 yield a qualitative insight into
phase transitions, the quantitative descrip-
tion of real materials needs a more detailed
analysis. In this section, we complement
the phenomenological theory by a more
microscopic approach. In the first step, the
essential degrees of freedom for a particu-
lar transition are identified and an appro-
priate model is constructed. In the second
step, the statistical mechanics of the model
are treated by suitable approximate or nu-
merical methods.
Fig. 4-12 illustrates the modeling of or-
der–disorder phase transitions in solids.
Transitions occur where the basic degree of
freedom is the (thermally activated) diffu-
sion process of atoms between various lat-
tice sites. This happens for unmixing alloys
such as Al–Zn or for ordering alloys such
as b-CuZn or the Cu–Au system. The mod-
eling of such systems will be discussed in
Sec. 4.3.1. Many structural transitions are
of a very different nature: we encounter pe-
riodic lattice distortions where atomic dis-
placements are comparable to those of lat-
tice vibrations. Short wavelength distor-
tions may give rise to “antiferrodistortive”
and “antiferroelectric” ordering, as exem-
plified by the perovskites SrTiO
3, PbZrO
3,
etc. Long wavelength distortions corre-
sponding to acoustic phonons give rise to
“ferroelastic” ordering. Sec. 4.3.2 will deal
with the mean field treatment of such phase
transitions where the order parameter is a
phonon normal coordinate, and Sec. 4.3.3
is devoted to numerical methods going be-
yond mean-field theory.
4.3.1 Models for Order–Disorder
Phenomena in Alloys
Since the diffusive motion of atoms in
substitutional alloys (Fig. 4-12) is so much
slower than other degrees of freedom (such
as lattice vibrations), we may describe the
configurational statistics of a substitutional
binary alloy, to which we restrict attention
for simplicity, by local occupation vari-
ables {c
i}. If lattice site i is taken by an
atom of species B, c
i=1, if it is taken by an
atom of species A, c
i=0.
Neglecting the coupling between these
variables and other degrees of freedom, the
Hamiltonian is then:
∫∫=
-
BB
AB
AA
0
2 1 4 109
11
+−
+− −
+− − − +…
≠
∑
ij
ij i j
ij ij
ij ij
cc
cc
cc
[()
()( )()
()()( )]
v
v
v
xx
xx
xxwww.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

336. where v
AA, v
AB, and v
BBare interactions
between pairs of AA, AB, and BB atoms.
In fact, terms involving three- and four-
body interactions may also occur, but are
not considered here. Also, effects due to
vacancies may easily be included but are
neglected here.
As is well known, Eq. (4-109) can be re-
duced to the Ising model, Eq. (4-56), by the
transformation S
i=1–2c
i= ± 1, apart from
a constant term which is of no interest to us
here. The “exchange interaction” J
ij
between spins iand jand “magnetic field”
Hin Eq. (4-56) are related to the interac-
tion parameters of Eq. (4-109) by
J(x
i–x
j)∫J
ij=[2v
AB(x
i–x
j) (4-110)
–v
AA(x
i–x
j)–v
BB(x
i–x
j)]/4
H
ij ij
ji=
-
AA BB
1
2
4 111
[( ) ( )]
()
()
vvxx xx−− −
⎛
⎝
⎜
⎞
⎠
⎟
−
≠
∑
Dm
where D mis the chemical potential differ-
ence between the two species.
The same mapping applies for the lat-
tice-gas model of fluids, which at the same
time can be considered as a model of ad-
sorbed layers on crystalline surface sub-
strates (in d= 2 dimensions) (see Binder
and Landau (1989) for a review) or as a
model of interstitial alloys, such as hydro-
gen or light atoms such as C, N and O in
metals (Alefeld, 1969; Wagner and Horner,
1974; Alefeld and Völkl, 1978). If we
interpret B in Eqs. (4-109)–(4-111) as an
occupied site and A as a vacant site, then
usually v
AB=v
AA= 0, such that 
(4-112)
ebeing the binding energy which a particle
feels at lattice site i(for adsorbates at sur-
faces, this is a binding to the substrate, and
a similar enthalpy term is expected when
∫∫=
0+−+
≠
∑∑
ij
ij i j i
i
cc cv()xx e
284 4 Statistical Theories of Phase Transitions
Figure 4-12.Degree of free-
dom essential for the descrip-
tion of various order–disor-
der transitions in solids. For
further explanations, see text.www.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

337. 4.3 Computational Methods Dealing with Statistical Mechanics 285
an interstitial is dissolved in a metal,
whereas for modeling of an ordinary gas–
liquid transition we would put 
e= 0). The
relationships analogous to Eqs. (4-110) and
(4-111) are then: 
(4-113)
J
ij=–v(x
i–x
j)/4 (4-114)
where 
mis the chemical potential and
v(x
i–x
j) the pairwise interaction of parti-
cles at lattice sites x
iand x
j.
Of course, it is straigthforward to gener-
alize this approach to binary (A–B) alloys
including as a third option that a lattice site
is vacant (V) or to ternary alloys; then
more complicated models result, e.g., the
Potts (1952) model, Eq. (4-20), or the
model of Blume et al. (1971). Another gen-
eralization occurs if one species (B) of a 
binary alloy is magnetic, e.g., Fe in Fe–Al
alloys (see Dünweg and Binder (1987)). 
In the latter case, the Hamiltonian is (the
true magnetic spin variable is now denoted
as 
s
i)
where J
m(x
i–x
j) denotes the strength of
the magnetic interaction, which we have
assumed to be of the same type as used in
the Heisenberg model (Eq. (4-102)) with
n= 3 and H
i=0.
We will not go further into the classifica-
tion of the various models here, but note
one general property of both models in
Eqs. (4-109) and (4-112) which becomes
evident when mapping the Ising Hamilton-
ian Eq. (4-56): for H= 0, this Hamiltonian
is invariant against a change of sign of all
spins. For H≠0, it is invariant against the
∫∫=-
BB m
AB
AA
0
4 115
21
11
()
{[()()]
()( )
()()( )}
+− −−⋅
+− −
+− − − +…
≠
∑
ij
ij ij ijij
ij ij
ij ij
cc J
cc
cc
v
v
v
xx xx
xx
xx ssss
H
ij
ji=−
[
++ −
]
≠
∑()/ ( )/
()
em24 vxx
transformation {S
i,H}Æ{–S
i,–H} . This
implies a particle–hole symmetry of the
phase diagram of the lattice gas model, or a
symmetry against interchange of A and B
in the phase diagram of the binary alloy
model. Even in very ideal cases, where
both partners A and B of a binary alloy
crystallize in the same structure and have
similar lattice spacings, the phase diagram
is in reality not symmetric around the line
·c
iÒ= 1/2 (see the Cu–Au phase diagram
reproduced in Fig. 4-13 as an experimental
example). There are various conceivable
reasons for this asymmetry of real phase
diagrams: (i) the assumption of strictly
pairwise potentials (Eq. (4-109)) fails, and
terms such as c
ic
jc
k, c
ic
jc
kc
lneed to be in-
cluded in the effective Hamiltonian; and
(ii) the potentials v
AA, v
AB, and v
BBin Eq.
(4-109) are not strictly constant but depend
Figure 4-13.Partial phase diagram of copper–gold
alloys in the temperature–concentration plane indi-
cating the existence regions of the three ordered
phases Cu
3Au, CuAu, and CuAu
3(cf. Fig. 4-8a and
c). These phases are separated from each other (and
from the disordered phase occurring at higher tem-
peratures) by two-phase coexistence regions. The
boundaries of these regions are indicated by full and
broken lines. Region II is a long-period modulated
version of the simple CuAu structure occurring in re-
gion I. Note that for strictly pairwise constant inter-
actions (of arbitrary range!) in a model such as Eq.
(4-109) the phase diagram should have mirror sym-
metry around the line c
Au= 50%. From Hansen
(1958).www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

338. on the average concentration ·cÒof the al-
loy, since the effective lattice parameter a
and the Fermi wavenumber k
Fand other
physical characteristics entering the effec-
tive potential of the alloy also depend 
on concentration ·cÒ. In fact, both reasons
probably contribute to some extent in real
alloys – the first-principles derivation of
model Hamiltonians such as Eq. (4-109)
from the electronic structure theory for sol-
ids is a challenging problem (Bieber and
Gautier 1984a,b; Zunger, 1994). Since the
effective interaction typically is oscillatory
in sign (Fig. 4-14) according to the Friedel
form
v(|x|)=Acos (2|k
F||x|+j)/|x|
3
(4-116)
where Aand 
jare constants, small changes
in |k
F|and/or the distances lead to rela-
tively large changes in v(|x|) for distant
neighbors. Whereas the interaction Eq. 
(4-116) results from the scattering of con-
duction electrons at concentration inhomo-
geneities, in other cases such as Hin metals
the effective interactions are due to the
elastic distortions produced by these inter-
stitials in their environment. The resulting
elastic interaction is also of long range
(Wagner and Horner, 1974).
Since the effective interactions that
should be used in Eqs. (4-109) to (4-115)
often cannot be predicted theoretically in a
reliable way, it may be desirable to extract
them from suitable experimental data. For
metallic alloys, such suitable experimental
data are the Cowley (1950) short-range or-
der parameters 
a(x)∫a(x
i–x
j) which de-
scribe the normalized concentration corre-
lation function, c ∫·c
iÒ=(1–M)/2, M∫·S
iÒ
(4-117)
which correspond to the normalized spin
correlation functions in the Ising spin rep-
a()
()
xx
ij
ij ij
cc c
cc
SS M
M
−≡
〈〉−
−
〈〉−
−
22
2
1 1
=
resentation (Eq. (4-117) for a ferromagnet is equivalent to the phenomenological Eq. (4-43)). In the disordered phase of an Ising spin system, it is straightforward to obtain the wavevector-dependent response func-
286 4 Statistical Theories of Phase Transitions
Figure 4-14.(a) Interaction potential V
lmn
=–2J (x)
(note that x=(l,m,n) –
a
2
, abeing the lattice spacing) vs.
distance |x|(measured as 
÷
----
l
2
+
--
m
2
-----
+n
2
) for NiCr
0.11
at T= 560 °C, as deduced from (b) short-range order
parameters 
a(x) measured by diffuse neutron scatter-
ing. The circles show the results of the high-temper-
ature approximation, Eq. (4-121), and crosses the re-
sult of the inverse Monte Carlo method. Full and
broken curves represent a potential function of the
Friedel form, Eq. (4-116), with different choices of
the Fermi wavevector k
Ffor ·100Òand ·110Òdirec-
tions, amplitude A and phase 
Fbeing treated as fit-
ting parameters. From Schweika and Hauboldt (1986).www.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

339. 4.3 Computational Methods Dealing with Statistical Mechanics 287
tion c(q) describing the response to a 
wavevector-dependent field (cf. Eqs. (4-32),
(4-34), and (4-40)), if we apply the molec-
ular field approximation (see the next sub-
section). The result is (Brout, 1965): 
(4-118)
where J
˜
(q) is the Fourier transform of the
“exchange interaction” J(x), 
(4-119)
In Eq. (4-118), we have once again invoked
that there is a “fluctuation relation” 
(= static limit of the so-called “fluctuation-
dissipation theorem”) relating 
c(q) to the
structure factor S(q) which is just the
Fourier transform of the correlations ap-
pearing in Eq. (4-117): 
(4-120)
Combining now Eqs. (4-118) and (4-120)
we see that the reciprocal of this diffuse
scattering intensity S(q) in q-space is sim-
ply related to the Fourier transformation of
the interactions as
(4-121)
For the model defined in Eqs. (4-109) to
(4-111), this expression even is exact in 
an expansion of 1/S(q) in a power series in
1/Tto leading order in 1/T, and is one of
the standard tools for inferring information
on interactions in alloys from diffuse scat-
tering data (Clapp and Moss, 1966, 1968;
Moss and Clapp, 1968; Krivoglaz, 1969;
Schweika, 1994). An example is shown 
in Fig. 4-14. Close to the order–disorder
phase transition, we expect corrections to
the molecular field expression Eq. (4-121)
due to the effects of statistical fluctuations
41
1
4
1
cc
S
J
kT
cc
()
()
˜
()
()
−
−−
q
q
=
B
SS SM
ji
ij ij
( ) exp [ ( )] ( )
()
qqrr=i
≠
∑ ⋅− 〈 〉−
2
˜
( ) ( ) exp [ ( )]
()
JJ
i
ji
jijqxxqxx=i
≠
∑ −⋅−
SkT
M
JkT M
() ()
[
˜
()/ ]( )
qq
q
==
B
Bc
1
11
2
2
−
−−
(cf. Sec. 4.2.2); a more accurate procedure to deduce J(x) from experimental data on
a(x) is the “inverse Monte Carlo method”
(Gerold and Kern, 1986, 1987; Schweika, 1989), but this method also relies on the as- sumption that an Ising model description as written in Eq. (4-109) is appropriate. On the level of the molecular field approxima- tion (for alloys this is usually referred to as the Bragg–Williams (1934) approxima- tion) or the Bethe (1935) approximation, it is possible to avoid models of the type of Eq. (4-109) and include the configurational degrees of freedom in an electronic struc- ture calculation (Kittler and Falicov, 1978, 1979). Although such an approach sounds very attractive in principle, the results are not so encouraging in practice, as shown in Fig. 4-15. The results of this method for Cu
3Au are compared with Monte Carlo data
of a nearest neighbor Ising model (Binder, 1980), with the cluster-variation treatment of the same model (Golosov et al., 1973) and with experimental data (Keating and Warren, 1951; Moss, 1964; Orr, 1960).
The conclusion of this subsection is that
the development of microscopic models for the description of order–disorder phenom- ena in alloys is still an active area of re- search, and is a complicated matter, be- cause the validity of the models can only be judged by comparing results drawn from the models with experimental data. How- ever, these results also depend on the ap- proximation involved in the statistical-me- chanical treatment of the models (e.g., the full and broken curves in Fig. 4-15 refer to the same nearest-neighbor Ising model,
whereas the dash-dotted curve refers to a different model). In the next subsections, we consider various sophistications of the statistical mechanics as applied to various models for phase-transition phenomena. More details about all these problems can be found in the Chapter by Inden (2001).www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

340. We have concentrated in this subsection
on alloys, but a similar discussion could
have also been presented for order –disor-
der phenomena in adsorbed monolayers
(see for example, Binder and Landau (1981,
1989), where the modeling of systems such
as H adsorbed on Pd(100) surfaces in terms
of lattice gas models is discussed), for mag-
netic transitions (De Jongh and Miedema,
1974), etc. Usually only the source of ex-
perimental data used to extract the interac-
tion parameters is different: e.g., for ferro-
magnets such as EuS, rather than using dif-
fuse magnetic neutron scattering in analogy
with Eq. (4-118) it is more convenient to
extract J
˜
(q) from the measurements of spin
wave dispersion curves found from inelas-
tic neutron scattering (Bohn et al., 1980).
4.3.2 Molecular Field Theory
and its Generalization
(Cluster Variation Method, etc.)
The molecular field approximation
(MFA) is the simplest theory for the de-
scription of phase transition in materials;
despite its shortcomings, it still finds wide-
spread application and has been described
in great detail in various textbooks (Brout,
1965; Smart, 1966). Therefore we do not
treat the MFA in full detail here, but rather
indicate only the spirit of the approach.
We start with the Ising ferromagnet, Eq.
(4-56). The exact Helmholtz energy can be
found formally from the minimum of the
functional (Morita, 1972)
(4-122)
where the sums extend over all configura-
tions of the spins in the system and P({S
i})
is the probability that a configuration {S
i}
occurs. Eq. (4-122) thus corresponds to the
thermodynamic relation F=U–TSwhere
the entropy Sis written in its statistical
interpretation. Minimizing Eq. (4-122) for-
mally with respect to Pyields the canonical
distribution
P
eq({S
i}) ~ exp[–∂
Ising({S
i})/k
BT]
as desired.
⎧∂=
Ising
B
{}
{}({ }) ({ })
({ }) ln ( { })
S
ii
S
ii
i
i
SPS
kT P S PS
=±
=±
∑
∑
+
1
1
288 4 Statistical Theories of Phase Transitions
Figure 4-15.(a) Order parameters for the A
3B struc-
ture (Fig. 4-8c) on the fcc lattice: Long-range order
parameter 
Y(LRO) and short-range order parameter
–
a
1for the nearest-neighbor distance (SRO) vs. tem-
perature, according to the Monte Carlo method 
(Binder, 1980), the cluster variation (CV) method in
the tetrahedron approximation (Golosov et al., 1973),
and the Kittler–Falicov (1978, 1979) theory. Data for
Cu
3Au after Keating and Warren (1951) (LRO) and
after Moss (1964) (SRO). (b) Ordering energy DU
(normalized to zero at T
cin the disordered state) vs.
temperature for the fcc A
3B alloy (top) and the fcc
AB alloy (bottom). Theoretical curves are from the
same sources as in (a); experimental data were taken
from Orr (1960) and Orr et al. (1960).www.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

341. 4.3 Computational Methods Dealing with Statistical Mechanics 289
The MFA can now be defined by factor-
izing the probability P({S
i}) of a spin con-
figuration of the whole lattice into a prod-
uct of single-site probabilities p
iwhich 
can take two values: p
+=(1+M )/2 is the
probability that the spin at site iis up 
and p
–=(1–M)/2 is the probability that it
is down, p
+–p
–=Mis the magnetization.
Now the expression J
ijS
iS
jp
ip
j(cf. Eq. 
(4-56)) summed over the possible values 
p
+and p
–simply yields J
ijM
2
, and hence
Eq. (4-122) reduces to, using Eq. (4-119)
Minimizing 
⎧
MFA
with respect to Mnow
yields the elementary self-consistency
equation:
(4-124)
As is well known, Eq. (4-124) implies a
second-order transition at T
c∫J
˜
(q=0)/k
B
with the same exponents as in the Landau
theory.
Clearly, in factorizing P({S
i}) into a
product of single-site probabilities and
solving only an effective single-site prob-
lem, we have disregarded correlation in the
probabilities of different sites. A system-
atic improvement is obtained if we approx-
imate the probability of configurations not
just by single-point probabilities but by us-
ing “cluster probabilities”. We consider
probabilities p
nc(k,i) that a configuration k
of the nspins in a cluster of geometric con-
figuration coccurs (c may be a nearest-
neighbor pair, or a triangle, tetrahedron, etc.).
Note k=1,…,2
n
for Ising spins whereas
k=1,…,q
n
for the q-state Potts model.
M
kT
Jq M H==
B
tanh [
˜
() ]
1
0+
11
2
02
1
2
1
2
1
2
1
2
2
N
Jq M HM
kT
MM
MM
⎧
MFA
B
= = (4-1 3)
˜
()
ln
ln
−
+
++ ⎛
⎝
⎞
⎠
⎡
⎣
⎢
+
−− ⎛
⎝
⎞
⎠
⎤
⎦
⎥
These probabilities can be expressed in
terms of the multi-spin correlation func-
tions g
nc(i)∫·S
iS
j1
…S
j
n
Ò, where the set of
vectors x
j1
–x
i,…,x
j
n
–x
idefines the n-
point cluster of type clocated at lattice site
i. The Helmholtz energy functional to be
minimized in this cluster variation method
(Kikuchi, 1951; Sanchez and De Fontaine,
1980, 1982; Finel, 1994) is a more compli-
cated approximation of Eq. (4-122) than
Eq. (4-123). If the largest cluster consid-
ered exceeds the interaction range, the en-
ergy term in F=U–TSis treated exactly;
unlike Eq. (4-123), the entropy is approxi-
mated. We find
where the coefficients 
g
ncare combinato-
rial factors depending on the lattice geome-
try and the clusters included in the approx-
imation (Kikuchi, 1951), e.g., in the tetra-
hedron approximation for the f.c.c. lattice,
the sum over cin Eq. (4-125) includes the
(nearest-neighbor) tetrahedron, the near-
est-neighbor triangle, the nearest-neighbor
pair, and the single site.
Assuming the ordered structure to be
known, the symmetry operations of the as-
sociated group can be applied to reduce 
the number of variational parameters in 
Eq. (4-125). In the MFA, there is a single
non-linear self-consistent equation (Eq. 
(4-124)) or a set of equations involving the
order-parameter components if a problem
more complicated than the Ising ferromag-
net is considered. In the CV method, a
much larger set of coupled non-linear
equations involving the short-range order
parameters g
nc(i) is obtained when we min-
imize Eq. (4-125). Therefore, whereas the
simple MFA is still manageable for a wide
⎧=
(4-1 5)
B
=
1
2
2
2
1
2
ij
ij
inc
nc
k
ncJikT
pki
j
n∑∑ ∑∑
∑ +
×
g
,
,
,()
(,)
r gwww.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

342. variety of systems (Brout, 1965; Smart,
1966), the CV method is essentially re-
stricted to Ising-type problems relevant for
phase transitions in metallic alloys (De
Fontaine, 1979; Finel, 1994). We discuss
the merits of the various approaches in the
next section.
At this point, we return to the formula-
tion of the MFA for the case of an arbitrary
type of ordering, rather than the simple
transition from paramagnetic to ferromag-
netic considered in the above treatment
(Eqs. (4-123) and (4-124)) of the Ising
model (Eq. (4-56)). Hence Eq. (4-123) can
obviously be generalized as follows (see
also De Fontaine, 1975):
where 
x
i=c
i–c=(M–S
i)/2, and constant
terms have been omitted. Expanding Eq.
(4-126) in terms of 
x
iyields, again omitting
a constant term,
with the coefficients
f
0¢¢(x≠0) = – 4J(x
i–x
j)
f
0¢¢(x=0)=k
BT/[c(1 –c)] (4-128)
and
f
0¢¢¢=–k
BT(2c– 1)/[c (1 –c)]
2
(4-129)
f
0
IV=2k
BT[3c
2
–3c+ 1)]/[c(1 –c)]
3
If we group the x
is properly into the sub-
lattices reflecting the (known or assumed)
state, Eq. (4-127) essentially yields the
Landau expansion in terms of the order-pa-
rameter components, as discussed in Sec.
4.2.1. Rather than defining sublattices a
D⎧
MFA
IV
=( 4-17
)
1
2
2
1
3
1
4
0
0
3
0
4
ij
ijij
i
i
i
if
ff
,
()
!!∑
∑∑′′−
+ ′′′+
xx xx
xx
⎧
MFA
B
=( 4-16)−−
++ −−
≠
∑
∑22
11
ij
ijij
i
ii i i
J
kT c c c c
()
[ln ( )ln( )]
xx xx
priori, it is often convenient to introduce
Fourier transformations
(4-130)
which yield:
where Gis a reciprocal lattice vector and
f˜¢¢(q) can be written as
f˜¢¢(q) = – 4J
˜
(q) + k
BT/[c(1 –c)] (4-132)
Comparing Eqs. (4-121) and (4-132), we
realize that the inverse structure factor (or
inverse “susceptibility”, 
c
–1
(q)) is simply
proportional to f˜¢¢(q), as it should be, since
X(q) and H (q) (apart from constants) are
canonically conjugate thermodynamic var-
iables. Of special importance now are the
points q
cwhere J
˜
(q) has its maximum and
correspondingly f˜¢¢(q) has a minimum, be-
cause for these wavevectors a sign change
of f˜¢¢(q) in Eq. (4-131) occurs first (at the
highest temperature). Hence a stability
limit of the disordered phase is predicted as
k
BT
c=4J
˜
(q
c)c(1 –c) (4-133)
If at this temperature a second-order transi-
tion occurs, it should actually be described
by a “concentration wave” X(q
c) as an or-
der parameter (see also the discussion 
following Eq. (4-31)). In many cases the
maxima of J
˜
(q) occur at special symme-
try points of the Brillouin zone, and sym-
metry considerations in reciprocal space
are useful for discussing the resulting or-
D| |⎧
MFA
IV
=
N
fX
N
fXXX
N
fXXXX
2
3
4
2
0
0
q
qq q
qq q q
qq
qq q
qq q G
qq q q∑
∑
∑′′
+ ′′′ ′ ′′
×+ ′+′′−
+ ′′′′′′
×
′′′
′ ′′ ′′′
˜
() ()
!
() ( ) ( )
()
!
() ( ) ( ) (
)
(
,,
,, ,
d
d
qqq q q G+′+′′+′′′−)(4-11
)3
X
N
i
i
i( ) exp [ ]qqx=i
1 x∑ −⋅
290 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

343. 4.3 Computational Methods Dealing with Statistical Mechanics 291
dered structures (De Fontaine, 1975, 1979;
Khachaturyan, 1973, 1983).
This formulation of order–disorder tran-
sitions in alloys is analogous to the treat-
ment of structural transitions, for which a
model Hamiltonian similar to Eq. (4-127)
can be written: rather than concentration
deviations 
x
iwe now have displacement
vectors u
l(x) associated with a lattice vec-
tor xfor the lth atomic species in the unit
cell. Just as it is useful to relate 
x
ito the
Fourier transform X(q) of the concentra-
tion deviation, it is useful to relate u
l(x) to
the phonon normal coordinate Q
k,l, de-
fined as 
(4-134)
where M
lis the mass of the atom of type l
at site R
i
lin the ith unit cell, e
l(k,x) is a
phonon polarization vector, 
llabels the
phonon branch and kits wavevector (Fig.
4-16). In this case, ·Q
k
0,l
0
Ò
Tplays the role
ux kxek
k
kl
l
lNM
Q( ) exp( ) ( , )
,
,
=i
1
l
l
l∑ ⋅
of an order-parameter component for the transition: in mean-field theory, the asso- ciate eigenfrequency vanishes at a temper- ature T
c(“soft phonon”). If this happens for
a phonon with wavevector k
0at the Bril-
louin zone edge, we have an antiferro- electric order, provided that the phonon is polar, i.e., it produces a local dipole mo- ment. For non-polar phonons, such as for the transition in SrTiO
3at T
c=106 K where
k
0=p(–
1
2
,–
1
2
,–
1
2
)/a(the soft phonon there
physically corresponds to an antiphase ro- tation of neighboring TiO
6octahedra, as
indicated schematically in Fig. 4-12), the transition leads to “antiferrodistortive” or- der. Long-wavelength distortions corre- sponding to optical phonons give rise  to ferroelectric ordering (an example is Pb
5Ge
3O
11, where k
0= 0 and T
c= 450 K
(Gebhardt and Krey, 1979)). There are also paraelectric–ferroelectric transitions of first order, e.g., the cubic–tetragonal tran- sition of BaTiO
3, such that no soft mode
occurs.
Just as the “macroscopic” ferroelectric
ordering can be associated with the normal coordinate Q
k
0=0,lof the associate optical
phonon as a microscopic order parameter characterizing the displacements on the atomic scale, the “macroscopic” ferroelas- tic orderings, where in the phenomenologi- cal theory, Eq. (4-27), a component of the strain tensor 
e
mnis used as an order pa-
rameter, can be related to acoustical pho- nons. Examples are the martensitic mate- rial In-25 at.% Tl where the combination c
11–c
12of the elastic constants nearly sof-
tens at T
c= 195 K, and LaP
5O
14where c
55
softens at T
c= 400 K and the structure
changes from orthorhombic to monoclinic (Gebhardt and Krey, 1979). Of course, this correspondence between the microscopic description of displacements in crystals in terms of phonons and the phenomenologi- cal macroscopic description in terms of po-
Figure 4-16.(a) Schematic temperature variation of
order parameter 
F=·Q
k
0,l
0
Ò
Tand square of the 
“soft-mode” frequency 
w
2
(k
0,l
0) at a displacive
structural transition. (b) Schematic phonon spectrum
of the solid at T>T
c. Note that either optical or
acoustic phonons may go soft, and for optical pho-
nons a softening may often occur at the boundary of
the first Brillouin zone rather than at its center.www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

344. larization fields, strain fields, etc., is a
well-known feature of condensed-matter
theory (Kittel, 1967).
As is well known, the Q
k,lare defined
such that the Hamiltonian of the crystal in
the quasi-harmonic approximation is diag-
onalized (Born and Huang, 1954):
Eq. (4-135) corresponds precisely to Eqs.
(4-127) and (4-131) if only the “harmonic
terms” (i.e., quadratic terms in the expan-
sion with respect to the 
x
is) are retained.
The vanishing of a soft mode, 
w
2
(k
0,l
0) ~
(T–T
c), is again equivalent to the vanish-
ing of a coefficient r~(T–T
c) of a quad-
ratic term in a Landau expansion. Of
course, as in the Landau theory, higher-or-
der “anharmonic” terms in Qare crucial for
the description of the ordered phase in
terms of a stable order parameter ·Q
k
0,l
0
Ò
for T<T
c.
These anharmonic terms thoroughly
modify the picture of the transition as ob-
tained from MFA, as they lead to a cou-
pling of the soft mode Q
k
0,l
0
with other
noncritical modes. This coupling among
modes gives rise to a damping of the soft
mode. In fact, under certain circumstances
even an overdamped soft mode and the ap-
pearance of a central peak are expected
(Gebhardt and Krey, 1979; Bruce and
Cowley, 1981).
A treatment of the anharmonic higher-
order terms of a Landau-like expansion 
in reciprocal space is cumbersome, as Eq.
(4-131) demonstrates. Denoting the ampli-
tude of the displacement vector produced
by the soft mode Q
k
0,l
0
in the unit cell ias
F
i, we may formulate a model for a one-
∫=
=(4-15)
UU
uu
UQ
ijll
i
l
j
l
lilj
0
2
0
22
1
2
1
2
3
+∂∂∂
×
+
′
′
′
∑
∑
,,, , ,
,
,
[/()()]
() ()
(, )
ab
a b
a b
l
l
wl
xx
xx
k
k
k
||
component structural transition as follows
(Bruce and Cowley, 1981):
(4-136)
where r, uand Care phenomenological co-
efficients. The last term corresponds to 
the harmonic interaction between displace-
ments in neighboring lattice cells, while
the anharmonicity has been restricted to the
“single-site Hamiltonian” [r
F
i
2/2 +uF
i
4/4].
Note that Eq. (4-136) is fully equivalent to
the “Hamiltonian” D
⎧
MFA
in Eq. (4-127) if
f
0¢¢¢= 0 is chosen, since (F
i–F
j)
2
=F
i
2+F
j
2
–2F
iF
j, and the coefficient Cis thus
equivalent to f
0¢¢(x
i–x
j) where x
i–x
jis a
nearest-neighbor distance, a. On the other
hand, the “
F
4
-model” in Eq. (4-136) can
be thought of as a lattice analog of the 
Helmholtz energy functional Eq. (4-10),
putting (
F
i–F
j)≈a∙— F
i(see Milchev
et al. (1986) for a discussion).
An interesting distinction concerns this
effective single-site Hamiltonian felt by the
atoms undergoing the distortion. We have
assumed that the ordered structure is
doubly degenerate; the atoms below T
ccan
sit in the right or the left minimum of a
double-well potential. If the single-site po-
tential above T
cis essentially of the same
type, and only the distribution of the atoms
over the minima is more or less random,
the transition is called “order–disorder
type”. This occurs, for example, for hydro-
gen-bonded ferroelectrics and is analogous
to the sublattice ordering described above
for alloys. On the other hand, if the single-
site potential itself changes above T
cto a
single-well form, the transition is called
“displacive”. Whereas it was often thought
that displacive structural transitions exhibit
well-defined soft phonons right up to their
transition temperature T
c, it has now be-
∫
F
FF
FF4
1
2
1
4
1
2
24
2
=
i
ii
ij
ij
ru
C∑
∑ +
⎡
⎣
⎢
⎤
⎦
⎥
+−
〈〉
()
,
292 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

345. 4.3 Computational Methods Dealing with Statistical Mechanics 293
come clear that all these structural transi-
tions with a one-component order parame-
ter acquire characteristics of order–disor-
der transitions close to T
c, as expected from
the “universality principle”, and therefore
the distinction between the character of a
structural transition as being “order–disor-
der” or “displacive” is not a sharp one 
(Bruce and Cowley, 1981).
It should also be noted that the softening
of 
w(k
0,l
0) near T
cdoes not mean that dis-
placements u
l(x) become very large. In
fact, the mean square displacement of an
atom at a structural transition is only ex-
pected to have an energy-like singularity 
(Meißner and Binder, 1975)
·u
l
2Ò
T–·u
l
2Ò
T
c
~(T/T
c–1)
1–a
(4-137)
where 
ais the specific heat exponent. This
is important because ·u
l
2Ò
Tis easily de-
duced experimentally from the Mössbauer
effect, from the Debye–Waller factor de-
scribing the temperature variation of Bragg
peaks in X-ray or neutron scattering, etc. 
We end this subsection with a comment
on the theory of first-order structural tran-
sitions. The common approach is to restrict
the analysis entirely to the framework of
the quasi-harmonic approximation, in
which the Helmholtz energy at volume V
and temperature Tis written as
Therefore, if the effective potentials speci-
fying the dynamical matrix ∂
2
U/[(∂x
i
l)
a
∂x
j
l)
b] in Eq. (4-135) are known, the pho-
non frequencies 
w
V(k,l) for a given vol-
ume and the free energy F(T,V) are ob-
tained. Of course in this approach, knowl-
edge of the structure of the material is 
assumed. First-order transitions between
FTV U TS
UV
kT kT
V
V
(, )
() (,)
ln [ exp( ( , )/ )]
,
,
=
=(4-18)
BB
−
+
+−−
∑
∑0
1
2
3
1
k
k
k
k
l
l
wl
wl∫
∫
different structures can be handled by per-
forming this calculation for both phases
and identifying the temperature T
cwhere
the free energy branches of the two phases
cross. Since the quasi-harmonic theory is a
calculation of the mean-field type, as
pointed out above, first order transitions
also show up via stability limits of the
phases, where the soft modes vanish; thus
we are not locating T
cbut rather tempera-
tures T
0or T
1(cf. Fig. 4-6b), which are 
often not very far from the actual transi-
tion temperature. This quasi-harmonic ap-
proach to structural phase transitions has
been tried for many materials. Typical ex-
amples include RbCaF
3(Boyer and Hardy,
1981) and the systems CaF
2and SrF
2
(Boyer, 1980, 1981a, b), which show phase
transitions to a superionic conducting state.
4.3.3 Computer Simulation Techniques
In a computer simulation, we consider a
finite system (e.g., a cubic box of size L
3
with periodic boundary conditions to avoid
surface effects) and obtain information on
the thermodynamic properties, correlation
functions, etc. of the system (as specified
by its model Hamiltonian) which is exact,
apart from statistical errors. However, this
approach is restricted to classical statistical
mechanics (including quasi-classical mod-
els such as Ising or Potts models), although
remarkable progress on application to
quantum problems has been made (Kalos,
1985; Kalos and Schmidt, 1984; De Raedt
and Lagendijk, 1985; Suzuki, 1992; Ceper-
ley, 1995). The principal approaches of this
type are the molecular dynamics (MD)
technique and the Monte Carlo (MC) tech-
nique (for reviews of MD, see Ciccotti
et al., 1987; Hoover, 1987; Hockney and
Eastwood, 1988; Binder and Ciccotti,
1996; of MC, see Binder, 1979, 1984a;
Mouritsen, 1984; Binder and Heermann,www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

346. 1988; Binder and Ciccotti 1996). In the
MD method, we numerically integrate New-
ton’s equation of motion which follows
from the chosen Hamiltonian, assuming er-
godic behavior, and the quantities of inter-
est are obtained as time averages from the
simulation. This method requires that the
relevant physical time scales involved in
the problem are not too different from each
other. The MD approach has often been ap-
plied successfully to liquid–solid transi-
tions. It would not be suitable to study or-
der–disorder phenomena in solid alloys,
since the time step in the MD method for a
solid must be much less than a phonon fre-
quency, and this time scale is orders of
magnitude smaller than the time between
diffusive hops of atoms to neighboring va-
cant sites, which is the process relevant to
equilibration of configurational degrees of
freedom (Fig. 4-12).
For problems of the latter type, the MC
method clearly is to be preferred. In the
MC method, random numbers are used to
construct a random walk through the con-
figuration space of the model system. Us-
ing the Hamiltonian, transition probabilities
between configurations are adjusted such
that configurations are visited according to
their proper statistical weight (Binder,
1979, 1984a; Binder and Ciccotti, 1996).
Again averages are obtained as “pseudo-
time averages” along the trajectory of the
system in phase space, the only difference
from the MD method being that the tra-
jectory is now stochastic rather than deter-
ministic. Both methods have been exten-
sively reviewed (see the references quoted
above); therefore, we do not give any de-
tails here, but only briefly mention the dif-
ficulties encountered when phase transi-
tions are studied, and discuss a few typical
examples of their application.
One principal difficulty is the finite-size
rounding and shifting of the transition. In
principle, this problem is well understood
(Fisher 1971; Challa et al., 1986; Binder,
1987b; Privman, 1990). In practice, this
makes it difficult to distinguish between
second-order and weakly first-order transi-
tions. For example, extensive MD work
was necessary to obtain evidence that the
melting transition of two-dimensional sol-
ids with pure Lennard–Jones interaction is
first order (Abraham, 1983, 1984; Bakker
et al., 1984), and that the suggested two
continuous transitions involving the hexa-
tic phase do not occur in these systems
(Nelson and Halperin, 1979). However,
this conclusion has been called into ques-
tion by recent simulations for hard-disk
fluids (Jaster, 1998) providing evidence for
continuous two-dimensional melting.
Another difficulty is that the periodic
boundary condition (for a chosen shape of
the box) prefers certain structures of a solid
and suppresses others which do not “fit”:
this is particularly cumbersome for incom-
mensurate modulated structures (Selke, 1988,
1989; 1992) and for off-lattice systems,
such as studies of the fluid–solid transition
or phase transitions between different lat-
tice symmetries. For example, particles in-
teracting with a screened Coulomb potential
(this is a model for colloidal suspensions or
colloidal crystals (see Alexander et al.,
1984)) may exhibit a fluid phase in addi-
tion to an f.c.c. and a b.c.c. crystal, and the
determination of a complete phase diagram
is correspondingly difficult (Kremer et al.,
1986, 1987; Robbins et al., 1988). The tra-
ditional approach to dealing with such prob-
lems is to repeat the calculation for differ-
ent box shapes and compare the free ener-
gies of the different phases. An interesting
alternative method has been proposed by
Parrinello and Rahman (1980) and Parri-
nello et al. (1983), who generalized the MD
method by including the linear dimensions
of the box as separate dynamic variables.
294 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

347. 4.3 Computational Methods Dealing with Statistical Mechanics 295
Another severe problem is the occur-
rence of metastability and hysteresis; the
system may become trapped in a meta-
stable state, the lifetime of which is longer
than the observation time of the simulation.
The distinction of such long-lived meta-
stable states from true equilibrium states is
difficult and may require computation of
the Helmholtz energies of the phases in
question.
These problems must be considered
when first-order transitions are studied by
computer simulation so that suitable box
sizes and observation times are chosen and
the initial state is prepared accordingly.
Then, employing sufficient effort in com-
puting time, and performing a careful anal-
ysis of all the possible pitfalls mentioned
above, very reliable and useful results can
be obtained which are superior in most
cases to any of the other methods that have
been discussed so far (see Fig. 4-15 for a
comparison in the case of f.c.c. alloys).
Misjudgements of the problems mentioned
above have also led to erroneous con-
clusions: in the f.c.c. Ising antiferromagnet
with nearest interaction J, the triple point
between the AB and A
3B structures
(Fig. 4-8a, c) was suggested to occur at
T= 0 (Binder, 1980) while later a nonzero
ut low temperature was found, k
BT/|J|~ 1.0
(Gahn, 1986; Diep et al., 1986; Kämmerer
et al., 1996). Unlike in the other methods,
such problems can always be clarified by
substantially increasing the computational
effort and carrying out more detailed anal-
yses of the simulation “data”.
A significant advantage of such a “com-
puter experiment” is that interaction pa-
rameters of a model can be varied system-
atically. As an example, Fig. 4-17 shows
the temperature dependence of the long-
range order parameters and some short-
range order parameters of an f.c.c model of
an A
3B alloy with nearest-neighbor repul-
sive (J
nn) and next-nearest-neighbor attrac-
tive interaction J
nnn(Eq. (4-110)), for vari-
ous choices of their ratio R=J
nnn/J
nn. The
comparison with experimental data shows
that a reasonable description of Cu
3Au is
obtained for R= – 0.2, for T<T
c, whereas
better experimental data for T>T
care
needed before a more refined fit of effec-
tive interaction parameters can be per-
formed.
As an example showing that the MC
method can deal with very complex phase
Figure 4-17.Temperature dependence of the long-
range order parameter 
Yof the Cu
3Au structure and
of the nearest- and next-nearest-neighbor short-range
order parameter 
a
1and a
2. Curves are Monte Carlo
results of Binder (1986). Points show experimental
data of Cowley (1950), Schwartz and Cohen (1965),
Moss (1964), Bardham and Cohen (1976), and Keat-
ing and Warren (1951). From Binder (1986).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

348. diagrams, we consider the model in Eq. 
(4-115) for alloys with one magnetic com-
ponent. However, in order to facilitate
comparison with the MFA and CV meth-
ods, the magnetic interaction is chosen to be
of Ising rather than Heisenberg type (Dün-
weg and Binder, 1987). Fig. 4-18 shows
the resulting phase diagrams for the choice
of interaction parameters J
nnn/J
nn= 0.5 and
J
m/|J
nn|= 0.7. In this case, the MFA pre-
dicts a wrong phase diagram “topology”
(e.g., a direct transition between paramag-
netic D0
3and A2 phases never occurs,
there is always a B2 phase in between), and
it also grossly overestimates the transition
temperatures. In contrast, the CV method
yields the phase diagram “topology” cor-
rectly, and it overestimates the transition
temperature by only a few per cent. Al-
though the CV method does not always
perform so well (see Fig. 4-15, and Binder
(1980) and Diep et al. (1986) for a discus-
sion in the case of f.c.c. alloys), it is always
much superior to the simple MFA, which in
many cases fails dramatically.
One great advantage of the MC method
over the CV method is that it can also be
applied straightforwardly to models with
continuous degrees of freedom, for which
the CV method would be cumbersome to
work out. As examples, Figs. 4-19 and 
4-20 show the phase diagrams of the model
in Eqs. (4-93) and (4-136), namely the an-
isotropic classical Heisenberg antiferro-
magnet in a uniform field H
||along the easy
axis and the 
F
4
model on the square lat-
tice. Note that in the latter case the result 
of the MFA would be off-scale in this 
figure (e.g., K
c
–1= 4 in the Ising limit). Fi-
nally, Fig. 4-21 shows the phase diagram of
a model for Eu
xSr
1–xS (Binder et al.,
1979):
(4-139)
∫=−⋅
≠
∑
1
2
ij
ij i j i j
JxxSS
296 4 Statistical Theories of Phase Transitions
Figure 4-18.Phase diagram of the body-centered
cubic binary alloy model with nearest and next near-
est neighbor crystallographic interactions J
nnand
J
nnn(both being “antiferromagnetic” if Ising-model
terminology is used) and a ferromagnetic nearest-
neighbor interaction J
mbetween one species, for
J
nnn/J
nn=0.5, J
m/|J
nn|= 0.7; cdenotes the concentra-
tion of the magnetic species. (a) Result of the MFA 
(Bragg–Williams approximation); (b) result of the
CV method in the tetrahedron approximation, and
part (c) the MC result. The A2 phase is the crystallo-
graphically disordered phase, the orderings of the B2
and D0
3phases are shown in Fig. 4-3. Two-phase co-
existence regions are shaded. The magnetic ordering
of the phases is indicated as para(-magnetic) and
ferro(-magnetic), respectively. From Dünweg and
Binder (1987).www.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

349. 4.3 Computational Methods Dealing with Statistical Mechanics 297
where S
iare unit vectors in the direction of
the Eu magnetic moment, and from spin
wave measurements (Bohn et al., 1980) it
is known that EuS essentially exhibits
superexchange between nearest and next-
nearest neighbors only, further-neighbor
interactions being negligibly small, with
J
nnn/ J
nn= – 1/2. Thus Fig. 4-21 exhibits a
remarkable agreement between computer
simulation and experiment (Maletta and
Felsch, 1979); there are no adjustable pa-
rameters involved whatsoever. Note that
analytical methods for such problems with
quenched disorder would be notoriously
difficult to apply.
To conclude this subsection, we state
that computer simulation techniques are
Figure 4-19.Phase diagram of a uniaxial classical
Heisenberg antiferromagnet on the simple cubic lat-
tice, as a function of temperature Tand field H
||ap-
plied in the direction of the easy axis. The anisotropy
parameter Din the Hamiltonian Eq. (4-93) is chosen
as D= 0.2. Both Monte Carlo results (crosses, circles,
full curves) and the Landau theory fitted to the phase
diagram in the region off the true bicritical point is
shown (dash-dotted straight line). The Landau theory
would overestimate the location of the bicritical tem-
perature T
b(T
b*>T
b), and fails to yield the singular
umbilicus shape of the phase diagram lines T
c
^, T
c
||
near the bicritical point. The broken straight lines de-
note the appropriate choices of “scaling axes” for a
crossover scaling analysis near T
b. The triangles
show another phase diagram, namely when the field
is oriented in the perpendicular direction to the easy
axis. The nature of the phases is denoted as AF (anti-
ferromagnetic), SF (spin-flop) and P (paramagnetic).
See Fig. 4-9a for a schematic explanation of this
phase diagram. From Landau and Binder (1978).
Figure 4-20.Critical line K
c
–1of the F
4
model 
(Eq. (4-136)) on the square lattice, shown in the space of couplings K
–1
, L
–1
defined as K=–rC/u,
L=r(r+4C)/(4u). Note that for L
–1
= 0, and Kfinite,
the square Ising model results, the transition temper- ature of which is exactly known (Onsager, 1944; ar- row). (∫) Monte Carlo simulation of Milchev et al.
(1986); (+) MD results of Schneider and Stoll (1976); (, ¥) from a real space renormalization
group calculation of Burkhardt and Kinzel (1979). The disordered phase occurs above the critical line. From Milchev et al. (1986).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

350. well suited for studying the phase diagrams
of various systems, magnetic systems, me-
tallic alloys, structural transitions, ad-
sorbed layers at surfaces, etc. provided that
a suitable model Hamiltonian is known.
Apart from the phase diagram, detailed in-
formation on long- and short-range order is
also accessible. Also, for certain problems,
the kinetics of phase transitions can be 
studied, in particular for alloys where the
Monte Carlo process can directly model
the atomic jump processes on the crystal
lattice (Binder, 1979, 1984a; Kehr et al.,
1989). Of course, there is also interest in
clarifying the phase diagrams of materials
in their fluid states. Computer simulations
can now obtain accurate gas–liquid phase
diagrams (Wilding, 1997), liquid binary
mixtures (Wilding et al., 1998) and surfac-
tants (Schmid, 1999).
4.4 Concepts About Metastability
Metastable phases are very common in
nature, and, for many practical purposes,
not at all distinct from stable phases (re-
member that diamond is only a metastable
modification of graphite!). Also, approxi-
mate theories of first-order phase transi-
tions easily yield Helmholtz energy
branches that do not correspond to the ther-
mal equilibrium states of minimum
Helmholtz energy, and hence are com-
monly interpreted as metastable or unstable
states (cf. Figs. 4-6 and 4-22). From the an-
alog of Eq. (4-12) for H(0,
(4-140)
the limit of metastability where 
c
T=
(∂
F/∂H)
Tdiverges as follows:
(4-141)
Since in the metastable states
c
T=(3k
BTu)
–1
(F
2
–F
s
2)
–1
we see that c
TÆ•as FÆF
s; in the
(T,
F) plane, the spinodal curve F=F
s(T)
plays the part of a line of critical points.
Similar behavior also occurs in many
other theories: for example, the van der
Waals equation of state describing gas–liq-
uid condensation exhibits an analogous
loop of one-phase states in the two-phase
coexistence region.
Unfortunately, although the description
of metastability in the framework of the
MFA seems straightforward, this is not so
if more accurate methods of statistical ther-
modynamics are used. A heuristic compu-
ru
kT
ru
H
rr
u
T
+
−
−
−
3
11
0
33
2
33
2
0
F
FF
s
B
s
c==
==
=
c
//
1
0
3
kTV
F
ru
H
kT
TBB
==
∂
∂
⎛
⎝
⎜
⎞
⎠ ⎟ +−F
FF
298 4 Statistical Theories of Phase Transitions
Figure 4-21.Ferromagnetic critical temperature of
the model Eq. (4-139) of Eu
xSr
1–xS vs. concentration
xof magnetic atoms. Circles denote experimental
data due to Maletta and Felsch (1979), triangles the
MC results for the diluted classical Heisenberg fcc
ferromagnet with nearest (J
nn) and next-nearest 
(J
nnn) exchange, J
nnn=––
1
2
J
nn. From Binder et al.
(1979).www.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

351. 4.4 Concepts About Metastability 299
tational approach, discussed in Sec. 4.3.2,
is the CV method where we systematically
improve upon the MFA by taking more and
more short-range correlations into account,
the larger the cluster is chosen. Computing
the equation of state for a nearest-neighbor
Ising ferromagnet in this way (Kikuchi,
1967), it is found that the stable branch is
nicely convergent, whereas the metastable
loop becomes flatter and flatter the larger
the cluster, i.e., the critical field H
c(Eq. 
(4-141)) converges towards zero. An exact
calculation of the equation of state yields
the magnetization jump from 
F
0to –F
0
as Hchanges from 0
+
to 0
–
in Fig. 4-22, but
does not yield any metastable state! This is
not surprising, however, because statistical
mechanics is constructed to yield informa-
tion only on thermal equilibrium states.
The partition function is dominated by the
system configurations in the vicinity of the
minimum of the Helmholtz energy. In a
magnet with H < 0, states with negative
magnetization have lower Helmholtz en-
ergy than those with positive magnetiza-
tion. Hence the latter do not result from the
partition function in the thermodynamic
limit. Similar conclusions emerge from a
rigorous treatment of the gas–fluid transi-
tion in systems with long-range interac-
tions (Lebowitz and Penrose, 1966).
Two concepts emerged for the descrip-
tion of metastable states (for a more de-
tailed discussion see Binder (1987a)). One
concept (Binder, 1973) suggests the defini-
tion of metastability be based on the con-
sideration of kinetics: starting with the sys-
tem in a state of stable equilibrium, the sys-
tem is brought out of equilibrium by a sud-
den change of external parameters (temper-
ature, pressure, fields, etc.). For example,
in the Ising-type ferromagnet of Fig. 4-22,
at times t< 0 we may assume we have a
state at H=0
+
, F=F
0> 0: at t= 0, the field
is switched to a negative value: then the
magnetization in equilibrium is negative,
but in the course of the nonequilibrium re-
laxation process from the initial state at 
F
0
a metastable state may occur with F
ms>0,
although H< 0, which exists only over a 
finite lifetime 
t
ms. In order that F
mscan
clearly be identified in this dynamic pro-
cess where the time-dependent order pa-
rameter 
F(t) relaxes from F(t=0) =F
0to
the negative equilibrium value 
F(tÆ•), 
it is necessary that 
t
msis much larger than
any intrinsic relaxation time. Then the time
t
msfor the decay of the metastable state far
exceeds the time needed for the system to
relax from the initial state towards the met-
astable state, and, in the “nonequilibrium
relaxation function” 
F(t) the metastable
state shows up as a long-lived flat part
where 
F(t)≈F
ms. The second concept
tries to define a metastable state in the
framework of equilibrium statistical me-
chanics by constraining the phase space so
as to forbid the two-phase configurations
Figure 4-22.Order parameter  Fvs. conjugate field
Haccording to the phenomenological Landau theory
for system at a temperature Tless than the critical
temperature T
cof a second-order phase transition
(schematic). At H= 0, a first-order transition from 
F
0
to –F
0occurs (thick straight line). The metastable
branches (dash-dotted) end at the “limit of meta-
stability” or “spinodal point” (
F
s, –H
s), respectively,
and are characterized by a positive-order parameter
susceptibility 
c
T> 0, whereas for the unstable branch
(broken curve) 
c
T<0.www.iran-mavad.com
+ s e l ≡'4 , kp e r i ≡&s ! 9 j+ N 0 e

352. that otherwise dominate the partition func-
tion. Langer (1974) suggested that phase
separation into two phases with order pa-
rameters –
F
0and +F
0coexisting at the
first-order transition at H= 0 in Fig. 4-1
and Fig. 4-22 is suppressed if we consider
a system (a) at fixed 
Fbetween –F
0and
+
F
0but (b) constrain the system and di-
vide it into cells of size L
d
, and (c) require
that the order parameter is not only glo-
bally fixed at 
Fbut also inside each cell. 
If Lis small enough, namely aOLO
x
coex
(this is the same condition as noted for 
the construction of a continuum model
from the microscopic Hamiltonian, see
Sec. 4.2.2, and Eqs. (4-9) and (4-57)),
phase separation inside a cell cannot occur,
and hence a coarse-grained Helmholtz en-
ergy density f
cg(F) of states with uniform
order parameter 
Fis obtained, which has
exactly the double-well shape of the Lan-
dau theory (Fig. 4-6a). Since f
cg(F) then
necessarily depends on this coarse-grained
length scale L, it is clear that the metastable
states of f
cg(F) are not precisely those 
observed in experiments, unlike the dy-
namic definition. While f
cg(F) defines a 
“limit of metastability” or “spinodal curve”
F
swhere (∂
2
f
cg(F)/∂F
2
)
T=c
T
–1vanishes
and changes sign, this limit of metastability
also depends on Land hence is not related
to any physical limit of metastability. In the
context of spinodal decomposition of mix-
tures, the problem of the extent to which a
spinodal curve is relevant also arises and is
discussed in detail in the chapter by Binder
and Fratzl (2001).
In some systems where a mean-field de-
scription is appropriate, the lifetime of
metastable states can be extremely long,
and then the mean-field concept of a limit
of metastability may be very useful. This
fact can again be understood in terms of 
the Ginzburg criterion concept as outlined
in Sec. 4.2.2. Since the spinodal curve
F=F
s(T) acts like a line of critical points,
as discussed above, we require (cf. Eq. 
(4-58))
·[
F(x)–F]
2
Ò
T,LO[F–F
s(T)]
2
(4-142)
i.e., the mean-square fluctuations in a vol-
ume L
d
must be much smaller than the
squared distance of the order parameter
from its value at criticality, which in this
case is the value at the spinodal. Now, the
maximum permissible value for Lis the
correlation length 
xin the metastable state,
which becomes
(4-143)
x={R/[6du F
s(T)]
1/2
}[F–F
s(T)]
–1/2
Eq. (4-143) exhibits the critical singularity
of 
xat F
s(T), corresponding to that of c
T
noted above. With a little algebra we find
from Eqs. (4-142) and (4-143) (Binder,
1984c)
This condition can only be fulfilled if the
interaction range is very large. Even then
for d= 3 this inequality is not fulfilled very
close to the spinodal curve.
In order to elucidate the physical signifi-
cance of this condition further and to dis-
cuss the problem of the lifetime of meta-
stable states, we briefly discuss the mean
field theory of nucleation phenomena. In
practice, for solid materials, heterogeneous
nucleation (nucleation at surfaces, grain
boundaries, dislocations, etc.) will domi-
nate; and this may restrict the existence of
metastable states much more than expected
from Eq. (4-144). However, we consider
here only the idealized case that homoge-
neous nucleation (formation of “droplets”
of the stable phase due to statistical fluctu-
ations) is the mechanism by which the met-
astable state decays. Since there is no
1
4
22 2
22 62
64
ORT
RT T
RH H
d
dd d
dd
x
−
−−
−
−
−
−
[()]
~ [ ()] [ ()]
~( )
()/ ()/
()/
FF
FFF
s
ss
c
(4-1 4
)
300 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

353. 4.4 Concepts About Metastability 301
mechanism by which this process can be
suppressed, metastable states cannot exist
in a region where strong homogeneous nu-
cleation occurs. Therefore we estimate the
intrinsic and ultimate limit of metastability
by the condition that the Helmholtz energy
barrier against homogeneous nucleation
decreases to the order of the thermal en-
ergy k
BT(Binder and Stauffer, 1976b).
As is well known, the nucleation barrier
arises from competition between the favor-
able) volume energy of a droplet of the new
phase and the (unfavorable) surface Helm-
holtz energy between the droplet and the
surrounding metastable background phase.
In order to calculate this Helmholtz energy
barrier in the framework of the Landau the-
ory, we can still use Eqs. (4-10) and (4-33).
Putting H(x) = 0 in Eq. (4-33) and consid-
ering the situation that 
F(zƕ)= F
0,
F(zÆ–•)=– F
0and solving for the con-
centration profile 
F(z) (Cahn and Hilliard,
1958), we can obtain the interface Helm-
holtz energy associated with a planar (infi-
nitely extended) interface perpendicular to
the z-direction (Fig. 4-23a). In the “classi-
cal theory of nucleation” (Zettlemoyer,
1969), this Helmholtz energy is then used
to estimate the Helmholtz energy barrier.
However, this “classical theory of nuclea-
tion” is expected to be reliable only for
metastable states near the coexistence
curve, where the radius R * of a critical
droplet (corresponding to a droplet Helm-
holtz energy exactly at the Helmholtz en-
ergy barrier DF*) is much larger than the
width of the interfacial profile (which then
is also of the same order as the correlation
length 
x
coexat the coexistence curve, cf.
Figs. 4-23a, b). Therefore it cannot be used
close to the limit of metastability. Cahn and
Hilliard (1959) have extended the Landau
theory to this problem, solving Eq. (4-33)
for a spherical geometry, where only a 
radial variation of 
F(r) with radius ris 
permitted, and a boundary condition
F(rƕ)=F
msis imposed (Fig. 4-
23b, c). Whereas for 
F
msnear F
coex=F
0
this treatment agrees with the “classical
theory of nucleation”, it differs signifi-
cantly from it for 
Fnear F
s(T): then the
critical droplet radius R* is of the same or-
der as the (nearly divergent!) correlation
length 
x(Eq. (4-143)), and the profile is
extremely flat, 
F(r)reaches in the droplet
center only a value slightly below 
F
srather
than the other branch of the coexistence
curve. Calculating the Helmholtz energy
Figure 4-23.Order-parameter profile  F(z) across
an interface between two coexisting phases ±
F
coex,
the interface being oriented perpendicular to the z-di-
rection (a) and the radial order-parameter profile for
a marginally stable droplet in a metastable state
which is close to the coexistence curve (b) or close to
the spinodal curve (c). In (a) and (b) the intrinsic
“thickness” of the interface is of the order of the cor-
relation length at coexistence 
x
coex, whereas in (c) it
is of the same order as the critical radius R*. From
Binder (1984c).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

354. barrier DF*, we obtain for Tnear T
c
(Binder, 1984c; Klein and Unger, 1983)
(4-145)
whereas near the coexistence curve the re-
sult is
(4-146)
where all prefactors of order unity are omit-
ted. In a system with a large range R of in-
teraction, the nucleation barrier is very high
in the mean-field critical region, in which
R
d
(1 –T/T
c)
(4 –d)/2
o1 (cf. Eq. (4-62)); this
factor, which controls the Ginzburg criter-
ion, also controls the scale of the nuclea-
tion barrier as a prefactor (see Fig. 4-24).
In this region the condition for the limit of
metastability, DF*/k
BT≈1, is located very
close to the mean-field spinodal. Then the
description of nucleation phenomena in
terms of the diffuse droplets described by
Fig. 4-23c near the spinodal curve is mean-
ingful (“spinodal nucleation”). On the
other hand, for a system with short-range
interactions where R(measured in units 
of the lattice spacing in Eqs. (4-145) and
(4-146)) is unity, the Helmholtz energy
barrier becomes of order unity long before
the spinodal curve is reached. The singu-
larity at the spinodal then completely lacks
any physical significance, as the meta-
stable state decays to the stable phase long
before the spinodal is reached.
It is instructive to compare the condition
DF*/k
BTO1 with Eq. (4-144): this shows
that these conditions are essentially iden-
tical! This is not surprising: the MFA is 
essentially correct as long as effects of 
statistical fluctuations are very small: the
“heterophase fluctuations” (droplets of the
DF
kT
R
T
T
d
dd*
~
()/ ()
Bc c
coex
coex
1
42 1
−
⎛
⎝
⎜
⎞
⎠
⎟
−⎛
⎝
⎜
⎞
⎠
⎟
−− −
FF
F
DF
kT
R
T
T
d
dd*
~
()/ ()/
Bc c
s
coex
1
42 62
−
⎛
⎝
⎜
⎞
⎠
⎟
−⎛
⎝
⎜
⎞
⎠
⎟
−−
FF
F
new phase) are also extremely rare, and the
Helmholtz energy cost to form them should
be very high, as implied by Eqs. (4-145)
and (4-146). 
These concepts have been tested by
Monte Carlo simulations on simple-cubic
Ising models with various choices of the
range of the interaction (Fig. 4-25, Heer-
mann et al., 1982). For the case of nearest-
neighbor interaction (each spin interacts
with q= 6 neighbors) it is seen that 
c
T
–1dif-
fers considerably from the MFA result, and
the range of fields over which metastable
states can be observed is not very large. With
an increase in the number of neighbors q,
302 4 Statistical Theories of Phase Transitions
Figure 4-24.Schematic plots of the Helmholtz en-
ergy barrier for (a) the mean field critical region, i.e.,
R
d
(1–T/T
c)
(4 –d)/2
o1, and (b) the non-mean field
critical region i.e., R
d
(1–T/T
c)
(4 –d)/2
O1. Note that
owing to large prefactors to the nucleation rate, the
constant of order unity where the gradual transition
from nucleation to spinodal decomposition occurs is
about 10
1
rather than 10
0
. From Binder (1984c).www.iran-mavad.com 
+ s e  l ≡'4 , kp e  r  i ≡&s ! 9 j+  N   0 e

355. 4.5 Discussion 303
this range increases, the value of c
T
–1at 
the field where nucleation becomes appre-
ciable falls also, and the 
c
T
–1vs. Hcurve
quickly converges towards the MFA pre-
diction, except in the immediate neighbor-
hood of the limit of metastability H
c.
It is expected that the ideas sketched
here will carry over to more realistic sys-
tems.
4.5 Discussion
There are many different kinds of phase
transitions in materials, and the fact that a
material can exist in several phases may
have a strong influence on certain physical
properties. The approach of statistical me-
chanics tries to provide general concepts
for dealing with such phenomena: classifi-
cation methods are developed which also
try to clarify which aspects of a phase 
transition are specific for a particular mate-
rial and which are general (“universal”). At
the same time, theoretical descriptions are
available both on a phenomenological
level, where thermodynamic potentials are
expanded in terms of suitable order param-
eters and the expansion coefficients are un-
determined, and can only be adjusted to ex-
perimental data, and on a microscopic
level, where we start from a model Hamil-
tonian which is treated either by molecular
field approximations or variants thereof or
by computer simulation techniques.
This chapter has not given full details of
all these approaches, but rather tried to
give a discussion which shows what these
methods can achieve, and to give the reader
a guide to more detailed literature on the
subject. An attempt has been made to sum-
marize the main ideas and concepts in the
field and to describe the general facts that
have been established, while actual materi-
als and their phase transitions are men-
tioned as illustrative examples only, with
no attempt at completenesss being made.
Although the statistical thermodynamics of
phase transitions has provided much physi-
cal insight and the merits and limitations of
the various theoretical approaches are now
well understood generally, the detailed
understanding of many materials is still ru-
dimentary in many cases; often there are
not enough experimental data on the order-
ing phenomenon in question, or the data
are not precise enough; a microscopic
model Hamiltonian describing the interac-
tion relevant for the considered ordering is
often not  explicitly known, or is so com-
plicated that a detailed theory based on
Figure 4-25.Inverse susceptibility of Ising ferro-
magnets plotted against h=–H/k
BTat T/T
c
MFA=4/9
for various ranges of the exchange interaction; each
spin interacts with equal strength with qneighbors.
The full curve is the MFA, the broken curve a fit to a
droplet model description. From Heermann et al.
(1982).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

356. such a model Hamiltonian does not exist.
Many of the more sophisticated methods
(CV method, computer simulation, etc.)
are restricted in practice to relatively sim-
ple models. Hence, there still remains a lot
to be done to improve our understanding of
phase transitions.
4.6 References
Abraham, F. F. (1974), Homogeneous Nucleation
Theory. New York: Academic.
Abraham, F. F. (1983), Phys. Rev. Lett. 50, 978.
Abraham, F. F. (1984), Phys. Rev. B29, 2606.
Aharony, A. (1973), Phys. Rev. B8, 1973.
Aharony, A. (1976), in: Phase Transitions and Criti-
cal Phenomena, Vol. 6: Domb, C., Green, M. S.
(Eds.). London: Academic, p. 358.
Alefeld, G. (1969), Phys. Stat. Sol. 32, 67.
Alefeld, G., Völkl, J. (Eds.) (1978), Hydrogen in
Metals I, II. Berlin: Springer Verlag (Topics in
Appl. Physics, Vols. 28, 29).
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308 4 Statistical Theories of Phase Transitionswww.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

361. 5 Homogeneous Second-Phase Precipitation
Richard Wagner
Forschungszentrum Jülich GmbH, Jülich, Germany
Reinhard Kampmann
Institut für Werkstofforschung, GKSS-Forschungszentrum GmbH, Geesthacht, Germany
Peter W. Voorhees
Department of Materials Science and Engineering, Northwestern University, Evanston,
Ill., USA
List of Symbols and Abbreviations  . . . . . . . . . . . . . . . . . . . . . . . . . . 311
5.1 Introduction................................. 314
5.2 General Considerations.......................... 315
5.2.1 General Course of an Isothermal Precipitation Reaction  . . . . . . . . . . 315
5.2.2 Thermodynamic Considerations – Metastability and Instability  . . . . . . 317
5.2.3 Decomposition Mechanisms: Nucleation and Growth versus Spinodal
Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
5.2.4 Thermodynamic Driving Forces for Phase Separation  . . . . . . . . . . . 322
5.3 Experimental Techniques for Studying Decomposition Kinetics.... 326
5.3.1 Microanalytical Tools  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
5.3.1.1 Direct Imaging Techniques  . . . . . . . . . . . . . . . . . . . . . . . . . 326
5.3.1.2 Scattering Techniques  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
5.3.2 Experimental Problems  . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
5.3.2.1 Influence of Quenching Rate on Kinetics  . . . . . . . . . . . . . . . . . . 330
5.3.2.2 Distinction of the Mode of Decomposition  . . . . . . . . . . . . . . . . . 332
5.4 Precipitate Morphologies.......................... 334
5.4.1 Experimental Results  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
5.4.2 Factors Controlling the Shapes and Morphologies of Precipitates  . . . . . 336
5.5 Early Stage Decomposition Kinetics.................... 339
5.5.1 Cluster-Kinetics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 340
5.5.1.1 Classical Nucleation – Sharp Interface Model . . . . . . . . . . . . . . . . 340
5.5.1.2 Time-Dependent Nucleation Rate  . . . . . . . . . . . . . . . . . . . . . . 343
5.5.1.3 Experimental Assessment of Classical Nucleation Theory  . . . . . . . . . 345
5.5.1.4 Non-Classical Nucleation – Diffuse Interface Model  . . . . . . . . . . . . 347
5.5.1.5 Distinction Between Classical and Non-Classical Nucleation . . . . . . . . 349
5.5.2 Diffusion-Controlled Growth of Nuclei from the Supersaturated Matrix  . . 350
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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362. 5.5.3 The Cluster-Dynamics Approach to Generalized Nucleation Theory  . . . . 352
5.5.4 Spinodal Theories  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
5.5.5 The Philosophy of Defining of ‘Spinodol Alloy’ – Morphologies of 
‘Spinodal Alloys’  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
5.5.6 Monte Carlo Studies  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
5.6 Coarsening of Precipitates......................... 370
5.6.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
5.6.2 The LSW Theory of Coarsening  . . . . . . . . . . . . . . . . . . . . . . . 370
5.6.3 Extensions of the Coarsening Theory to Finite Precipitate Volume Fractions  373
5.6.4 Other Approaches Towards Coarsening  . . . . . . . . . . . . . . . . . . . 377
5.6.5 Influence of Coherency Strains on the Mechanism and Kinetics of 
Coarsening – Particle Splitting  . . . . . . . . . . . . . . . . . . . . . . . 377
5.7 Numerical Approaches Treating Nucleation, Growth and Coarsening as
Concomitant Processes........................... 381
5.7.1 General Remarks on the Interpretation of Experimental Kinetic Data 
of Early Decomposition Stages   . . . . . . . . . . . . . . . . . . . . . . . 381
5.7.2 The Langer and Schwartz Theory (LS Model) and its Modification by 
Kampmann and Wagner (MLS Model)  . . . . . . . . . . . . . . . . . . . 383
5.7.3 The Numerical Modell (N Model) of Kampmann and Wagner (KW)  . . . . 385
5.7.4 Decomposition of a Homogeneous Solid Solution  . . . . . . . . . . . . . 385
5.7.4.1 General Course of Decomposition  . . . . . . . . . . . . . . . . . . . . . . 385
5.7.4.2 Comparison Between the MLS Model and the N Model  . . . . . . . . . . 387
5.7.4.3 The Appearance and Experimental Identification of the Growth 
and Coarsening Stages  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
5.7.4.4 Extraction of the Interfacial Energy and the Diffusion Constant from
Experimental Data  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
5.7.5 Decomposition Kinetics in Alloys Pre-Decomposed During Quenching  . . 391
5.7.6 Influence of the Loss of Particle Coherency on the Precipitation Kinetics  . 392
5.7.7 Combined Cluster-Dynamic and Deterministic Description 
of Decomposition Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 394
5.8 Self-Similarity, Dynamical Scaling and Power-Law Approximations. . 395
5.8.1 Dynamical Scaling  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
5.8.2 Power-Law Approximations  . . . . . . . . . . . . . . . . . . . . . . . . . 398
5.9 Non-Isothermal Precipitation Reactions................. 401
5.10Acknowledgements............................. 402
5.11References.................................. 402
310 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
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363. List of Symbols and Abbreviations 311
List of Symbols and Abbreviations
a lattice parameter
A solvent atoms
a
i
j(c) activity of atomic species iin phase jat composition c
B solute atoms
c composition
c
A atomic fraction of A
c
0 initial composition
c
r
mean composition
Dc supersaturation or composition difference
c
a
e equilibrium composition of matrix phase a
c¢
a
e composition of matrix ain metastable equilibrium with coherent precipi-
tates
c
b
e equilibrium composition of precipitate phase b
c
p composition of incoherent particles
c¢
p composition of coherent particles
c
ij elastic constant
c
R composition of matrix at the matrix/particle interface
C(i) equilibrium cluster distribution
D diffusion constant
D
–
mean diameter of precipitated particles
E Young’s modulus
f(R) size distribution function of precipitates with radii R
F(c),f(c) Helmholtz energy or energy density
F¢(c),f¢(c) Helmholtz constraint energy or constraint energy density
DF* nucleation barrier
DF
el elastic free energy
DF
a/b interfacial free energy
DF
ch(c) chemical driving force
f
p precipitated volume fraction
F˜(x) time-independent scaling function
G Gibbs energy
G(|r–r
0|) two-point correlation function at spatial positions r, r
0
H enthalpy
i, i* number of atoms in a cluster or in a cluster of critical size
J
s
, J* steady state and time-dependent nucleation rate
K aspect ratio of an ellipsoid of revolution
K* gradient energy coefficient in the CH spinodal theory
K
R
LSW coarsening rate according to the LSW theory
k Boltzmann constant
L ratio of elastic to interfacial energy
M atomic mobility
N
v number density of precipitates
n
v number of atoms per unit volumewww.iran-mavad.com
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364. p pressure
R radius of precipitates
R
–
mean radius of precipitates
R* radius of a critical nucleus or a particle being in unstable equilibrium
with the matrix
R
g molar gas constant
R(
k); R(l) amplification factor in the CH spinodal theory
S entropy
S(
k) structure function
S
m=S(k
m) maximum of structure function
T temperature
T
A annealing temperature
T
H homogenization temperature
t aging time
U internal energy
V volume
V
a, V
b molar volume of aor bphase
Z Zeldovich factor
a, a¢ matrix phase
b equilibrium precipitated phase
b¢ metastable precipitated phase; transition phase
g shear strain
d misfit parameter
h= (1/a
0)(∂a/∂c) atomic size factor
k, k scattering vector and its magnitude
k
m wavenumber of the maximum of the structure function S( k)
l wavelength
m shear modulus
m
i
j chemical potential of component iin phase j
n Poisson’s ratio
W atomic volume
s
ab specific interfacial energy
t incubation period or scaled time in the MLS and KW models
AEM analytical transmission electron microscopy
AFIM analytical field ion microscopy (atom probe field ion microscope)
CH Cahn–Hilliard spinodal theory
CHC Cahn–Hilliard–Cook spinodal theory
CTEM conventional transmission electron microscopy
EDX energy-dispersive X-ray analysis
EELS electron energy loss spectroscopy
FIM field ion microscopy
HREM high resolution electron microscopy
KW Kampmann–Wagner model
312 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

365. List of Symbols and Abbreviations 313
LBM Langer–Bar-On–Miller non-linear spinodal theory
LS Langer–Schwartz theory
LSW Lifshitz–Slyozov–Wagner theory
MCS Monte Carlo simulation
MLS modified Langer–Schwartz model
n.g. nucleation and growth
SAXS small angle X-ray scattering
SANS small angle neutron scattering
s.d. spinodal decomposition
TAP tomographic atom probewww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

366. 5.1 Introduction
Many technologically important proper-
ties of alloys, such as their mechanical
strength and toughness, creep and corro-
sion resistance, and magnetic and super-
conducting properties, are essentially con-
trolled by the presence of precipitated par-
ticles of a second phase. This commonly
results from the decomposition of a solid
solution during cooling. A fundamental
understanding of the thermodynamics, the
mechanism and the kinetics of precipita-
tion reactions in metallic solids, leading to
a well-defined microstructure, is therefore
of great interest in materials science.
As is reflected by the schematic binary
phase diagram of Fig. 5-1, for reasons of
entropy the single-phase state aof a solid
solution with composition c
0is thermody-
namically stable only at elevated tempera-
tures. At lower temperatures the free en-
ergy of the system is lowered through un-
mixing (‘decomposition’ or ‘phase separa-
tion’) of a into two phases, a¢and b.
In order to initiate a precipitation reac-
tion, the alloy is first homogenized in the
single-phase region at T
Hand then either
a) cooled down slowly into the two-phase
region a¢ + b, or
b) quenched into brine prior to isothermal
aging at a temperature T
Awithin the
two-phase region (Fig. 5-1).
In both cases, thermodynamic equilib-
rium is reached if the supersaturation Dc,
defined as 
Dc(t)=c¯–c
a
e (5-1)
becomes zero. (Here c¯(t) is the mean matrix
composition at time twith c¯(t=0)∫ c
0.) 
For case a), which frequently prevails
during industrial processing, the aging
temperature and the associated equilibrium
solubility limit c
a
e(T) decrease continu-
ously. Equilibrium can only be reached if
the cooling rate is sufficiently low within a
temperature range where the diffusion of
the solute atoms is still adequately high.
The precipitated volume fraction (f
p) and
the dispersion of the particles of the second
phase can thus be controlled via the cool-
ing rate.
Procedure b) is frequently used for stud-
ies of decomposition kinetics under condi-
tions which are easier to control and de-
scribe theoretically (T =T
A= const.; D=
const.) than for case a). This leads to pre-
cipitate microstructures whose volume
fraction and particle dispersion depend on
Dc(T) and the aging time t.
Decomposition reactions involve diffu-
sion of the atomic species via the vacancy
and/or the interstitial mechanism. Hence,
the precipitate microstructure proceed-
ing towards thermodynamic equilibrium
evolves as a function of both time and tem-
perature. In practice, a metallurgist is often
requested to tailor an alloy with a specific
precipitation microstructure. For this pur-
314 5 Homogeneous Second-Phase Precipitation
Figure 5-1.Schematic phase diagram of a binary al-
loy displaying a miscibility gap. Dashed lines show
the metastable coherent solvus line and a possible
metastable intermetallic phase b¢. The long arrow in-
dicates the quenching process.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

367. 5.2 General Considerations 315
pose it would be desirable to have a theo-
retical concept, preferentially available as
a computer algorithm, that allows suitable
processing parameters to be derived for es-
tablishing the specific microstructure on
grounds of the given thermodynamics of
the alloy. This less empirical approach to
alloy design, however, would require a
comprehensive theory of precipitation ki-
netics, which has not yet been developed to
a satisfactory level, despite recent efforts
and progress made in elucidating the kinet-
ics of first-order phase transitions in solids.
Furthermore, in materials science it is fre-
quently desirable to predict the kinetic evo-
lution of an originally optimized precipitate
microstructure under service conditions,
e.g., for high-temperature applications in
two-phase materials, where the precipitate
distribution might undergo changes be-
cause of coarsening. Even though the ki-
netics of coarsening are of great practi-
cal importance, a completely satisfactory
coarsening theory has so far only been de-
veloped in the limit of zero precipitated
volume fraction (see Sec. 5.6). This limit,
however, is never realized in technical al-
loys, where the volume fraction of the mi-
nor phase frequently exceeds 30%.
In the present chapter recent theoretical
and experimental studies on the kinetics of
phase separation in solids are reviewed from
the point of view of the experimentalist.
Special emphasis is placed on the questions
to what extent the theoretical results can be
verified experimentally and to what extent
they might be of practical use to the physical
metallurgist. We confine ourselves to deal-
ing only with homogeneous and continuous
phase separation mechanisms. Heterogene-
ous nucleation at crystal defects, and dis-
continuous precipitation reactions at mov-
ing interfaces, as well as unmixing in solids
under irradiation are treated separately, in
the chapter by Purdy and Bréchet (2001).5.2 General Considerations
5.2.1 General Course of an Isothermal
Precipitation Reaction
As illustrated for a Ni–37 at.% Cu–8
at.% Al alloy (Fig. 5-2 and 5-3), an isother-
mal precipitation reaction (the kinetics of
which will be dealt with in more detail in
Secs. 5.5, 5.6 and 5.7) is qualitatively char-
acterized by an early stage during which an
increasing number density
N
v(t)=Úf(R)dR (5-2)
of more or less spherical solute-rich clus-
ters (‘particles’) with a size distribution 
f[R(t)] and a mean radius
(5-3)
are formed. In common with all homogene-
ous precipitation reactions studied so far, in
the earliest stages of the reaction the parent
phase aand the precipitate phase bshare a
common crystal lattice, i.e., the two phases
are coherent. As inferred from the field ion
micrographs of Fig. 5.2a–c and from the
quantitative data of Fig. 5-3, during the
early stages R
–
increases somewhat and the
supersaturation, Dc, decreases slowly. This
small reduction in Dc, however, is sufficient
to terminate the nucleation of new particles,
as indicated by the maximum of N
v(t). 
Beyond this maximum the precipitate
number density decreases (Fig. 5-3) due to
the onset of the coarsening reaction, during
which the smaller particles redissolve, thus
enabling the larger ones to grow (see Sec.
5.6). During this coarsening process the
supersaturation Dcdecreases asymptoti-
cally towards zero.
As mentioned previously, within the mis-
cibility gap, the solid solution a becomes un-
stable and decomposes into the stable solid
solution a¢and the precipitated phase b :
aÆa¢+b
R
Rf R R
fR R
=
∫
∫
()
()
d
dwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

368. During the first step of the precipitation
sequence a metastable precipitate (transi-
tion phase b¢) is formed, frequently with
large associated coherency strains and a
small interfacial energy 
s
ab. Often the
metastable phase b¢is an intermetallic
compound with a lower solute concentra-
tion (c¢
b
e∫c¢
p) than the equilibrium precipi-
tate (c
p). Because of the large coherency
strains, the metastable (‘coherent’) solvus
line (dashed line in Fig. 5-1) is shifted to-
wards higher solute concentrations. This
leads to a reduction of the supersaturation
316 5 Homogeneous Second-Phase Precipitation
Figure 5-2.a)–c) Neon field ion image of  g¢-precipitates (bright images in dark matrix) in Ni–36.8 at.% Cu–8
at.% Al aged for the given times at 580°C (Liu and Wagner, 1984). d) Three-dimensional distribution of Mg and
Zn atoms in a commercial Al–5.5 at.% Mg–1.35 at.% Zn alloy analyzed with the tomographic atom probe. Each
point represents an atom. The T¢precipitates are enriched in Mg and Zn (Bigot et al., 1997).
The terminal solute concentration of the
solvent (A)-rich matrix a¢is given by the
equilibrium solubility limit c
a
e(T); the ter-
minal composition c
pof the precipitated
phase is given by the solubility at T
Aat the
B-rich side of the phase diagram (Fig. 5-1).
Frequently, the interfacial energy 
s
ab
(J/m
2
) between the matrix and the equilib-
rium precipitate is rather high, particularly
if the width of the miscibility gap is large
(see Sec. 5.7.4.4). In this case, the decom-
position of aproceeds via the sequence
aÆa≤+b¢Æa¢+b
a) 2 min.; 2R
–
= 2,4 nm b) 180 min.; 2R
–
=8nm
c) 420 min.; 2R
–
=11nm d)www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

369. 5.2 General Considerations 317
with respect to the (incoherent) equilib-
rium solvus line (Fig. 5-1), and, hence, to a
reduction of the driving force for precipita-
tion (see Sec. 5.2.4). Often only after ex-
tended aging do the metastable phases a≤
and b¢transform further to the final equi-
librium phases a¢ and b. The crystal struc-
ture of the equilibrium phase bfinally pre-
cipitated is different from that of the parent
phase, leading to either a coherent, a semi-
coherent or a fully incoherent a¢/binterface
boundary (Gleiter, 1983). The atomic struc-
ture of the latter resembles that of a high-
energy, high-angle grain boundary and, thus,
is associated with a rather large interfacial
energy and small elastic strain energies.
The decomposition of Cu–Ti alloys with
Ti contents between ≈1 at.% and 5 at.%
serves as an example of such a complex pre-
cipitation sequence (Wagner et al., 1988): 
a-Cu–(1...5 at.%) Ti
1st stepT≈350 °C
a≤-Cu–Ti (f.c.c. solid solution) +
b¢-Cu
4Ti (metastable, coherent, body-
centered-tetragonal structure,
large coherency strains, small
interfacial energy,
s
ab= 0.067 J/m
2
)
2nd step  extended
aging
at T>350°C
a¢-Cu–Ti (f.c.c. solid solution) +
b-Cu
3Ti (stable, hexagonal structure,
incoherent, small strain, large
interfacial energy,
s
ab> 0.6 J/m
2
)
5.2.2 Thermodynamic Considerations –
Metastability and Instability
Let us consider a binary alloy consisting
of N
Asolvent atoms A and N
Bsolute atoms
B with N
A+ N
B=N, or, in terms of atomic
fractions, c
A= N
A/Nand c
B= N
B/N, with
c
A+ c
B= 1. (The concentrations of atoms
A and B are then given as c
An
vand c
Bn
v,
where n
vis the number of atoms per unit
volume.) As only one independent variable
remains, we can refer to the composition of
the alloy as c∫c
B(0≤c≤1).
Decomposition of a supersaturated sin-
gle-phase alloy into a two-phase state com-
monly occurs at constant temperature T
and pressure p, and is thus prompted by a
possible reduction in Gibbs energy (Gas-
kell, 1983)
G=H–TS (5-4)
Figure 5-3.Time evolution
of the mean radius R
–
, the
number density N
v, and 
the supersaturation Dcof
g¢-precipitates in Ni–36
at.% Cu–9 at.% Al during
aging at 500°C (Liu and
Wagner, 1984).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

370. with the enthalpy
H=U+pV (5-5)
Thermodynamic equilibrium is attained
when Ghas reached a minimum, i.e.,
dG
T, p=dU+pdV–TdS= 0 (5-6)
For phase separation in solids the term pdV
can usually be neglected with respect to the
others in Eq. (5-6). Thus a good approxi-
mation for Gis given by the Helmholtz en-
ergy
F=U–TS (5-7)
which will be used in the following sec-
tions as the relevant thermodynamic func-
tion. Equilibrium is achieved if F or the
corresponding Helmholtz energy density
(Fper unit volume or per mole) are mini-
mized. Unmixing only takes place if the
transition from the single-phase state low-
ers the Helmholtz energy, i.e., by conven-
tion if DF<0.
For a given temperature, volume and so-
lute concentration of a heterogeneous bi-
nary alloy, equilibrium between the two
phases aand bcan only be achieved if the
concentrations of A and B in the two
phases have been established such that
(5-8)
In other words, at equilibrium the chemical
potentials of the
component i(either A or B) in the two
phases are identical and the two phases
have a common tangent to the associated
free energy curves. This fact is illustrated
in Fig. 5-4 for a supersaturated solid solu-
tion of composition c
0, which decomposes
into a B-depleted phases a¢and a B-rich
phase bof composition c
a
eandc
b
e, respec-
tively. c
a
eand c
b
eare fixed by the common
mm
i
i
i
i
F
n
F
n
α
α
β
β
=
∂
∂
=
∂
∂
and
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
=
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
F
n
F
n
i
TVn
i
TVn
j j
α β
,, ,,
tangent to the Helmholtz energy curves of
the a- and b-phases; thus c
b
e–c
a
erepresents
the width of the miscibility gap at a given
temperature (cf. Fig. 5.4).
Fig. 5-5a shows schematically the phase
diagram of a binary alloy with a two-phase
region at lower temperatures for T=T
1; the
associated Helmholtz energy versus com-
position curve, F(c), is shown in Fig. 5-5b.
In the single-phase field a, Finitially de-
creases with increasing solute concentra-
tion due to the growing entropy of mixing
(Eq. (5-7)). In the thermodynamically
equilibrated two-phase region, Fvaries lin-
early with c(bold straight line satisfying
the equilibrium condition 
m
i
a=m
i
b).
The mean field theories (see Secs. 5.5.1
and 5.5.4) dealing with the unmixing kinet-
ics of solid solutions quenched into the
miscibility gap are now based on the (ques-
tionable) assumption that the quenched-in
single-phase states within the two-phase
region can be described by a ‘constraint’
Helmholtz energy F′(c)>F(c), e.g., as is
318 5 Homogeneous Second-Phase Precipitation
Figure 5-4.Helmholtz energy Fas a function of
composition for a binary alloy with a miscibility gap.
The changes in Fand the resulting driving forces for
unmixing are illustrated.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

371. 5.2 General Considerations 319
shown in Fig. 5-5b (bold dashed line).
Based on this concept, the miscibility gap
can be subdivided into a metastable region,
where F′(c)>F(c) and ∂
2
F′/∂c
2
> 0, and
an unstable region, for which F′(c)>F(c)
but ∂
2
F′/∂c
2
< 0. The unique spinodal
curve, which is defined as the locus of the
inflection points (∂
2
F′/∂c
2
)
T∫0, separ-
ates the two regions (Fig. 5-5a). The es-
sence of the distinction between meta-
stability and instability will be discussed in
the following section.
5.2.3 Decomposition Mechanisms:
Nucleation and Growth versus Spinodal
Decomposition
Experimentalists are often inclined to
distinguish between two different kinds of
decomposition reactions, depending on
whether i) the solid solution experiences a
shallow quench (e.g., from points 0 to 1,
Fig. 5-5a) into the metastable region or ii)
whether it is quenched deeply into the un-
stable region of the miscibility gap (e.g.,
from points 0 to 2).
For case i, unmixing is initiated via the
formation of energetically stable solute-
rich clusters (‘nuclei’). As inferred from
Fig.5-5 b) only thermal composition fluctu-
ations with sufficiently large compositional
amplitudes ·c–c
0Òlower (DF< 0) the
Helmholtz energy of the system and,
hence, can lead to the formation of stable
nuclei. According to the tangent construc-
tion for the given composition c
0of the in-
itial solid solution, the largest decrease in
DFis obtained for a nucleus with composi-
tion c
p
nuc(which depends on c
0) rather than
with c
p=c
b
e. The latter is the composition
of the terminal second phase bcoexisting
in equilibrium with the aphase of compo-
sition c
a
e. The formation of stable nuclei via
localized ‘heterophase’ thermal composi-
tion fluctuations (Fig. 5-6 a) requires a nu-
cleation barrier (typically larger than 5kT)
to be overcome (see Sec. 5.5.1) and is char-
acterized by an incubation period. This de-
fines the homogeneous solid solution at
point 1 (Fig. 5-5 a) as being metastable,
and the transformation as a nucleation and
growthreaction.
Figure 5-5.a) Phase diagram of a binary model al-
loy with constituents A and B. The two-phase region
is subdivided by the mean-field spinodal curve into
metastable (hatched) and unstable regions (cross-
hatched). b) Schematic Helmholtz energy versus
composition curves at temperature T
1. The bold
dashed curve in the two-phase region shows the 
‘constraint’ Helmholtz energy F¢(c) of the unstable
solid solution.www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

372. As will be outlined in Sec. 5.5.1, the de-
cay of a metastable solid solution via nu-
cleation and growth has frequently been
described in terms of cluster kinetics mod-
els. The cluster kinetics approaches are es-
sentially based on the Becker–Döring the-
ory (Becker and Döring, 1935) of the dy-
namics of solute cluster formation. There it
is assumed that the non-equilibrium system
consists of non-interacting solute-rich clus-
ters of various size embedded in the matrix.
The time evolution of the cluster size dis-
tribution and, hence, the dynamics of the
decay of the metastable alloy, are assumed
to proceed via the condensation or evapo-
ration of single solute atoms at each clus-
ter.
In case ii, the non-equilibrium solid so-
lution with an initial composition c
0>c
a
s,
(Fig. 5-5 b) is unstable with respect to the
formation of non-localized, spatially ex-
tended thermal composition fluctuations
with small amplitudes. Hence, the unmix-
ing reaction of an unstable solid solution,
which is termed spinodal decomposition, is
initiated via the spontaneous formation 
and subsequent growth of coherent 
(‘homophase’) composition fluctuations
(Fig. 5-6b).
The dynamic behavior of an unstable al-
loy proceeding towards equilibrium has of-
ten been theoretically approached in terms
of spinodal theories (see Sec. 5.5.4 and
Chapter 6), amongst which the most well-
known linear theory is due to Cahn (1966)
and the most elaborate non-linear one is
due to Langer et al. (1975). As discussed in
Sec. 5.2.2, the spinodal theories are based
on the assumption that in each stage of the
decomposition reaction the Helmholtz en-
ery of the non-equilibrium solid solution,
which contains compositional fluctuations,
can be defined. As in the cluster kinetics
models, the driving force for phase separa-
tion is again provided by lowering the
Helmholtz energy of the alloy. If the form
of the ‘constraint’ Helmholtz energy F
¢(c)
of the non-uniform system is properly cho-
sen, the composition profile with asso-
ciated minimum Helmholtz energy can be
determined (in principle!) at any instant of
the phase transformation.
The existence of a unique spinodal curve
within the framework of the mean field
theories (Cahn, 1966; Cook, 1970; Skripov
and Skripov, 1979) has led to the idea (still
320 5 Homogeneous Second-Phase Precipitation
Figure 5-6.Spatial variation in solute distribution
c(r=(x,y,z)) during a) a nucleation and growth re-
action, and b) a continuous spinodal reaction at the
beginning (time t
1) and towards the end (t
3) of the
unmixing reaction. The notation of the compositions
refers to Fig. 5-5; c
0
msand c
0
usare the nominal com-
positions of the quenched-in metastable and unstable
solid solutions, respectively; R* is the critical radius
of the nuclei and 
lthe wavelength of the composition
fluctuations. The direction of the solute flux is indi-
cated by the arrows. After an extended reaction time
(e.g., after t
3), the transformation products are simi-
lar and do not allow any conclusions to be drawn
with respect to the early decomposition mode.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

373. 5.2 General Considerations 321
widespread in the community of metallur-
gists) that there is a discontinuity of the
mechanism and, in particular, of the de-
composition kinetics at the boundary
between the metastable and unstable re-
gions. Therefore, many experiments have
been carried out in order to determine the
spinodal curve and to search for a kinetic
distinction between metastable and un-
stable states (see Sec. 5.5.5).
In reality, there is no need to develop dy-
namical concepts which are confined to ei-
ther the metastable nucleation and growth
regime (case i) or to the unstable spinodal
regime (case ii). In fact, the cluster-kinetics
models and the spinodal theories can be
seen as two different approaches used to
describe phase separation, the dynamics of
which are controlled by the same mecha-
nism. i.e. diffusion of solvent and solute at-
oms driven by the gradient of the chemical
potential (Martin, 1978).
This fact is reflected in the attempts to
develop ‘unified theories’which comprise
spinodal decomposition as well as nuclea-
tion and growth. Langer et al. (1975) tried
to develop such a theory on the basis of a
non-linear spinodal theory (see Sec. 5.5.4),
whereas Binder and coworkers (Binder et
al., 1978; Mirold and Binder, 1977) chose
the cluster kinetics approach by treating spi-
nodal decomposition in the form of a gener-
alized nucleation theory (see Sec. 5.5.3).
These theories involve several assumptions
whose validity is difficult to assess a priori.
The quality of all spinodal or cluster kinetics
concepts, however, can be scrutinized by
Monte Carlo simulations of the unmixing
kinetics of binary ‘model’ alloys. These are
quenched into either a metastable or an un-
stable state (see Sec. 5.5.6) and can be de-
scribed in terms of an Ising model (Kalos et
al., 1978; Penrose et al., 1978).
Although both the ‘unified theories’ and
the Monte Carlo simulations only provide
qualitative insight into early-stage unmix-
ing behavior of a binary alloy, they have re-
vealed that there is no discontinuity in the
decomposition kinetics to be expected dur-
ing crossing of the mean-field spinodal
curve by either increasing the concentra-
tion of the alloy and keeping the reaction
temperature (T
A) constant or by lowering
the reaction temperature and keeping the
composition (c
0) constant. On the other
hand, the mean-field description is strictly
valid only for systems with infinitely long-
range interaction forces (Gunton, 1984)
and, hence, in general does not apply to
metallic alloys (polymer mixtures might be
close to the mean-field limit: Binder, 1983,
1984; Izumitani and Hashimoto, 1985; see
Sec. 5.5.4). Therefore, numerous experi-
ments that have been designed by metallur-
gists in order to determine a unique spino-
dal curve simply by searching for drastic
changes in the dynamic behavior of an al-
loy quenched into the vicinity of the mean-
field spinodal must be considered with
some reservations.
In principle, the above-mentioned theo-
ries and, in particular, the Monte Carlo
simulations deal mainly with the dynamic
evolution of a two-phase mixture in its
early stages. They frequently do not ac-
count for a further evolution of the precipi-
tate or cluster size distribution with aging
time (i.e., coarsening; see Sec. 5.6) once
the precipitated volume fraction is close to
its equilibrium value. On the other hand,
the time evolution of the precipitate micro-
structure beyond its initial clustering stages
has been the subject of many experimental
studies and is of major interest in practical
metallurgy. For this purpose, numerical ap-
proaches have been devised (Langer and
Schwartz, 1980; Kampmann and Wagner,
1984) which treat nucleation, growth and
coarsening as concomitant processes and
thus allow the dynamic evolution of thewww.iran-mavad.com
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374. two-phase microstructure to be computed
during the entire course of a precipitation
reaction (see Sec. 5.7).
As was pointed out in Sec 5.2.1, the re-
action path of a supersaturated solid solu-
tion can be rather complex, sometimes in-
volving the formation of one or more inter-
mediate non-equilibrium phases prior to
reaching the equilibrium two-phase micro-
structure. Unlike in the ‘early stage
theories’ mentioned above, these compli-
cations, which are of practical relevance,
can be taken into consideration in numeri-
cal approaches. Even though they still con-
tain a few shortcomings, numerical ap-
proaches lead to a practical description of
the kinetic course of a precipitation reac-
tion which lies closest to reality.
There are several comprehensive review
articles and books dealing in a more gen-
eral manner with the kinetics of first-order
phase transitions (Gunton and Droz, 1984;
Gunton et al., 1983; Binder, 1987; Gunton,
1984; Penrose and Lebowitz, 1979). Phase
separation in solids (crystalline and amor-
phous alloys, polymer blends, oxides and
oxide glasses) via homogeneous nucleation
and growth or via spinodal decomposition
represents only one aspect among many
others (Gunton and Droz, 1984). Apart
from the above-mentioned numerical ap-
proaches, the comprehensive articles by
Martin (1978) and by de Fontaine (1982)
cover most of the theoretical developments
relevant to the kinetics of (homogeneous)
phase separation in metallic systems. A
general overview of the broad field of dif-
fusive phase transformations in materials
science including heterogeneous nuclea-
tion and discontinuous precipitation not
covered in the present chapter, can be
found in the article by Doherty (1983) or,
as an introduction to this field, in the book
by Christian (1975), see also chapter by
Purdy and Bréchet (2001).
5.2.4 Thermodynamic Driving Forces
for Phase Separation
Even in the single-phase equilibrium
state, the mobility of the solvent and solute
atoms at elevated temperatures permits the
formation of composition fluctuations
which grow and decay again reversibly
with time. If the solid solution is quenched
into the miscibility gap the two-phase mix-
ture is the more stable state and, thus, some
of these fluctuations grow irreversibly ow-
ing to the associated reduction in Helm-
holtz energy. The reduction in Helmholtz
energy during the transformation from the
initial to the final state provides the driving
force DF. As we shall see in Sec. 5.5.1, it is
possible to calculate the formation rate and
the size of stable composition fluctuations
(‘nuclei’ of the second phase) by means of
the cluster kinetics approach once DFis
known. It should, however, be emphasized
at this point that for most alloys it is rather
difficult to calculate DF with sufficient ac-
curacy. This is seen as one of the major
hindrances to performing a quantitative
comparison between theory and experi-
ment.
The driving force for precipitation is
made up of two different contributions: 
i) the gain in chemical Helmholtz en-
ergy, DF
ch< 0, associated with the forma-
tion of a unit volume of the precipitating
phase b, and 
ii) the expenditure of distortion Helm-
holtz energy, DF
el> 0, accounting for the
coherency strains which result from a
likely variation of the lattice parameter
with the spatial composition fluctuations.
i) Chemical contribution, DF
ch. Accord-
ing to the tangent rule and referring to Fig.
5-4, the chemical driving force for precipi-
tation of the equilibrium bphase out of a
solid solution with composition c
0is given
by the numerical value of xy
––
. Assuming the
322 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
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375. 5.2 General Considerations 323
precipitating phase balready has the final
bulk composition c
b
e∫c
prather than c
p
nuc
(which is only a reasonable assumption if
the supersaturation is not too large (Cahn
and Hilliard, 1959a, b)), DF
chhas been de-
rived for a unit volume of bphase with mo-
lar volume V
bas (Aaronson et al., 1970 a,
b):
(5-9)
where a
i(c) is the activity of the solvent 
(i= A) or solute component (i= B) in the
parent phase afor the given composition.
For most alloy systems the activity data
required for a computation of DF
chby
means of Eq. (5-9) are not available. In
principle, they can be derived from a com-
putation of the thermodynamic functions
by means of the CALPHAD method. Orig-
inally this method was developed for the
calculation of phasediagrams by Kaufman
and coworkers (Kaufman and Bernstein,
1970) on the basis of a few accurately
measured thermodynamic data to which
suitable expressions for the thermodynam-
ic functions had been fitted. For the sake of
a simplified mathematical description, the
stable solid solutions of the particular alloy
system are frequently described in terms of
the regular solution model, whereas the
phase fields of the intermetallic com-
pounds are approximated by line com-
pounds. The thermodynamic functions ob-
tained thereby are then used to reconstruct
the phase diagram (or part of it). The de-
gree of self-consistency between the recon-
structed and the experimentally determined
phase diagram (or the agreement between
the measured thermodynamic data and the
derived data) serves as a measure of the ac-
curacy of the thermodynamic functions.
DF
RT
V
c
ac
ac
c
ac
ac
ch
g
e B
B
e
e A
A
e=
×+
⎡
⎣
⎢
⎤
⎦
⎥
–
ln
()
()
(– )ln
()
()
β
β
α
β
α
00
1
Hence, if, for example, F(c,T) is known
for the stable solid solution, this value can
readily be extrapolated into the adjacent
two-phase region to yield F¢(c) (see Fig.
5-5) of the supersaturated homogeneous
solid solution or the related chemical po-
tentials and activities. Of course, physi-
cally this is only meaningful if we assume
(as, in fact is done for the derivation of Eq.
(5-9)) that the free energy of the non-equi-
librium solid solution can be properly de-
fined within the miscibility gap.
For many binary alloys and for some ter-
nary alloys of technological significance,
the thermodynamic functions have been
evaluated by means of the CALPHAD
method and are compiled in volumes of the
CALPHAD series (Kaufman, 1977).
Hitherto, due to the lack of available ac-
tivity data, the activities entering Eq. (5-9)
were frequently replaced by concentra-
tions, i.e.,
(5-10)
If bis almost pure B, then c
b
e≈1 and the
chemical driving force is approximated as
(5-11)
For many alloys the condition c
b
e≈1 is
notmet. Nevertheless, it has been used for
computation of DF. As discussed by Aa-
ronson et al. (1970 a), the resulting error
can be rather large. If the nucleating phase
is a solvent-rich intermetallic compound
(e.g., Ni
3Al or Cu
4Ti), the computation of
DF
chis particularly difficult, since up to
now the corresponding activities are nei-
ther known nor furnished by the CAL-
PHAD method, and Eqs. (5-10) and (5-11)
are no longer valid.
DF
RT
V
c
c
ch
g
e≈–lnβ α
0
DF
RT
V
c
c
c
c
c
c
ch
g
e
e
e
e=
×+
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
–
ln ( – ) ln
–
–
β
β
α
β
α
00
1
1
1www.iran-mavad.com
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376. ii) Reduction of the driving force by the
elastic strain energy, DF
el. The formation
of a composition fluctuation is associated
with the expenditure of elastic strain Helm-
holtz energy, DF
el, if the solvent and the so-
lute atoms have different atomic radii. Ac-
cording to Cahn (1962), the Helmholtz en-
ergy of a system containing homophase
fluctuations is raised by 
DF
el=h
2
Dc
2
Y (5-12)
where 
h= (1/a
0)(∂a/∂c) denotes the change
in lattice parameter (a
0for the homogene-
ous solid solution of composition c
0) with
composition, and Dcthe composition am-
plitude. Yis a combination of elastic con-
stants and depends on the crystallographic
direction of the composition modulation. It
reduces to E/(1 –v), Eand vbeing Young’s
modulus and Poisson’s ratio, respectively,
if the elastic anisotropy A∫2c
44+c
12–
c
11is zero; otherwise, in order to minimize
DF
el, the composition fluctuations are ex-
pected to grow along the elastically soft di-
rections, which for cubic crystals and A >0
are the ·100Òdirections. 
In the context of heterophase fluctuations,
the barrier against nucleation of the new
phase is dominated by the matrix/ nucleus
interfacial energy (see Sec. 5.5.1). Since co-
herent interfaces have a lower energy than
incoherent ones, a precipitate is usually co-
herent (or at least semi-coherent) during the
early stages of nucleation and growth. Often
its lattice parameter, a
b, is slightly different
from that of the parent phase, a
a. The result-
ing misfit between the (unstrained) matrix
and the (unstrained) precipitate:
(5-13)
can be accommodated by an elastic strain 
if both 
dand the particles are sufficiently
small; this is commonly the case during the
early stages of nucleation and growth.
d=
+
2
aa
aa
α β
α
β
–
The problem of calculating DF
elfor co-
herent inclusions is rather complex. It has generally been treated within the frame- work of isotropic elastic theory for ellip- soidal precipitates with varying axial ra- tios, e.g. Eshelby (1957), and has been ex- tended to the anisotropic case (Lee and Johnson, 1982). Basically, these treatments show that the strain energy depends on the particular shape of the new phase. Only if the transformation strains are purely dilata- tional and if aand bhave about the same
elastic constants will DF
elbecome inde-
pendent of the shape. It is then given as (Eshelby, 1957):
(5-14)
with 
gbeing the shear modulus.
As is shown schematically in Fig. 5-7,
with respect to the Helmholtz energy of the incoherent equilibrium phase b, the Helm-
holtz energy of the coherent phase b¢is
raised by the elastic energy to match the two lattices. The driving force DF
inc
∫xy
––
DF
el=
+⎛
⎝
⎞
⎠
2
1
1
2
g
n
n
d
–
324 5 Homogeneous Second-Phase Precipitation
Figure 5-7.Helmholtz energy curves and associated
driving forces for precipitates of the coherent meta-
stable (F
b¢) and incoherent equilibrium (F
b) phases.www.iran-mavad.com
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377. 5.2 General Considerations 325
is reduced to the value of DF
coh
∫uv
––
, and
the solubility limit in the aphase increases
from c
a
eto c¢
a
e.
Hence, if we refer to the resulting coher-
ent phase diagram (cf. Fig. 5-1), the total
driving force, D F, for coherent unmixing,
implicitly already containing the elastic
contribution, is obtained by replacing, e.g.,
in Eq. (5-11), the composition c
a
eof the in-
coherent phase by the composition c¢
a
eof
the coherent one:
(5-15)
As an example, for metastable iron-rich
f.c.c. precipitates (c
b
e≈99.9 at.%) with
d=–8¥10
–3
formed at 500 °C in a super-
saturated Cu–1.15 at.% Fe alloy (c
0= 1.15
at.%, (c
a
e≈0,05 at.%; Kampmann and Wag-
ner, 1986), Eq. (5-12) yields DF
el≈0.13
kJ/mol. This value is negligibly small with
respect to the rather large chemical driving
force (≈20 kJ/mol). Since, however, DF
ch
decreases markedly with increasing tem-
perature (or decreasing supersaturation),
whereas DF
eldoes not, coherent nucleation
commonly occurs at larger undercoolings
or supersaturations, whereas nucleation of
incoherent precipitates takes place at
smaller ones. This is in fact observed in the
Cu–Ti (see Sec. 5.2.1) and Al–Cu (Hornbo-
gen, 1967) systems.
As will be shown in Sec. 5.5.1.1, the bar-
rier DF* against formation of the new
phase, and hence the nucleation rate, is not
only a function of the driving force but is
rather sensitive to the nucleus/matrix inter-
face energy 
s
ab. Therefore, the phase nu-
cleating first will not necessarily be the
equilibrium phase with the lowest Helm-
holtz energy but that with the lowest DF*,
e.g. a coherent metastable phase with a 
low value of 
s
ab. This explains the likely
formation of a series of metastable phases
(see Sec. 5.2.1) in various decomposing al-
D
a
F
RT
V
c
c
=
′
′
–ln
g
e
β
0
loys, amongst which Al–Cu (c
0≤2.5 at.%
Cu) is probably the best known. There the stable phase (q-CuAl
2) is incoherent, with
a high associated 
s
ab(R1 J/m
2
) (Hornbo-
gen, 1967). This inhibits homogeneous nu- cleation and q is found to form only at
small undercoolings, preferentially at grain boundaries (see Fig. 5-8). At larger under- coolings, a series of metastable copper-rich precipitates is formed in the order: GPI zones 
ÆGPII zones (q≤)  Æq¢with differ-
ent crystal structures (see Fig. 5-8). Gui- nier–Preston zones of type I (GPI) and type II (GPII or q≤) are coherent and nucleate homogeneously, whereas q¢is semi-coher-
ent and nucleates preferentially at disloca- tions.
Al–Mg–Si alloys represent an important
group of age-hardenable structural materi- als heavily used by industry in both cast and wrought form. Hence, control and opti- mization of the precipitate microstructure during cooling and heat treatment is of par-
Figure 5-8.Solubility limits of Cu in Al in the pres-
ence of the metastable q≤and q¢phases (dashed) and
the stable qphase, as a function of temperature (after
Hornbogen, 1967).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

378. amount interest for optimum mechanical
properties (Bratland et al., 1997). How-
ever, due to its inherent complexity, the
precipitation sequence of these commercial
alloys is still a matter of controversy (Dutta
and Allen, 1991; Gupta and Lloyd, 1992;
Edwards et al., 1998).
5.3 Experimental Techniques for
Studying Decomposition Kinetics
5.3.1 Microanalytical Tools
In general, the course of a decomposi-
tion reaction, including the early stages
(during which composition fluctuations
and second-phase nuclei are formed, see
Sec. 5.2.1) and the coarsening stages, can-
not be followed continuously by any one
microanalytical technique. The progress of
the reaction is usually reconstructed from
the microstructure that develops at various
stages of the phase transformation. Thus it
is necessary to analyze the spatial exten-
sion and the amplitude of composition
fluctuations of incipient second-phase par-
ticles, as well as the morphology, number
density, size and chemical composition of
individual precipitates at various stages of
the phase transformation. For this purpose
microanalytical tools are required that are
capable of resolving very small (typically a
few nm) solute clusters, and which allow
(frequently simultaneously) an analysis of
their chemical composition to be made.
The tools that, in principle, meet these
requirements can be subdivided into two
groups: direct imaging techniques and
scattering techniques.
It is beyond the scope of this chapter to
discuss any one of the techniques belong-
ing to either group in any detail. We shall
only briefly summarize the merits and the
shortcomings of the various techniques
with respect to both the detection limit and
the spatial resolution of microanalysis.
5.3.1.1 Direct Imaging Techniques
Field ion microscopy (FIM) (Wagner,
1982) as well as conventional (CTEM) 
(Hobbs et al., 1986) and high resolution 
(HREM) transmission electron microscopy
(Smith, 1983) allow for direct imaging of
the second-phase particles, provided the
contrast between precipitate and matrix is
sufficient.
In CTEM both the bright field and the
dark field contrast of particles less than 
≈5 nm in diameter are often either too
weak or too blurred for an accurate quanti-
tative determination of the relevant structu-
ral precipitate parameters. Hence, CTEM
does not provide access to an experimental
investigation of the early stages of decom-
position but remains a technique for study-
ing the later stages. In contrast, HREM al-
lows solute clusters of less than 1 nm diam-
eter to be imaged, as was demonstrated for
Ni
3Al precipitates in Ni–12 at.% (Si-Qun
Xiao, 1989) and for silver-rich particles in
Al–1 at.% Ag (Ernst and Haasen, 1988;
Ernst et al., 1987). As is shown in Fig. 5-9
for the latter alloy, the particles are not
ideally spherical but show some irregular-
ities which, however, are small in compari-
son with their overall dimensions. Hence,
describing their shapes by spheres of radius
Ris still a rather good approximation.
Prior to measuring the particle sizes di-
rectly from the HREM micrographs, it
must be established via computer image
simulations that there exists a one-to-one
correspondence between the width of the
precipitate contrast (dark area in Fig. 5-9)
and its true size. This was verified for the
HREM imaging conditions used in Fig.
5.9. The evolution of the size distribution
in Al–1 at.% Ag with aging time at 413 K
326 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
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379. 5.3 Experimental Techniques for Studying Decomposition Kinetics 327
as derived from HREM micrographs is
shown in Fig. 5-10.
The HREM lattice imaging technique
has also been employed to determine the
spacings of adjacent lattice planes in vari-
ous alloys undergoing phase separation
(Sinclair and Thomas, 1974; Gronsky et
al., 1975). The smooth variations observed
were attributed to variations in lattice pa-
rameter caused by composition modula-
tions such as we would expect from spino-
dal decomposition (e.g. Fig. 5-6 b at t
1or
t
2). Subsequent model calculations of high-
resolution images, however, showed that
the spatial modulation of the lattice fringes
observed on the HREM image can be sig-
nificantly different from the spatial mod-
ulation of the lattice plane spacings in the
specimen (Cockayne and Gronsky, 1981).
In practice, it does not appear possible to
derive reliable information on composition
modulations from modulations of lattice
fringe spacings in HREM images.
The composition of the imaged particles
can, in principle, be obtained from a nalyti-
cal electron microscopy (AEM), which is
routinely based on energy dispersive X-ray
analysis (EDX) or, less frequently, on e lec-
tron energy loss spectroscopy (EELS)
(Williams and Carter, 1996). Due to beam
spreading by electron beam/specimen
interaction, the spatial resolution for rou-
tine EDX microanalysis is limited to 
R10 nm. This value is rather large and con-
fines EDX microanalysis of unmixing al-
loys to the later stages of precipitation. The
more interesting early stages of decom-
position, however, where composition
changes of either extended or localized so-
lute fluctuations are expected to occur, are
not amenable to EDX microanalysis.
The spatial resolution of EELS may,
with difficulty, reach 1 nm (Williams and
Carter, 1996), but so far, EELS has not
been employed for a systematic microanal-
Figure 5-9.HREM lattice fringe image of Al-1 at.%
Ag aged for 92 h at 413 K with silver-rich precipi-
tates (dark areas). For imaging the Al matrix a rather
large defocus (280 nm) had to be chosen, which
gives rise to the bright Fresnel fringes surrounding
the particles (reproduced by courtesy of F. Ernst 
(Ernst and Haasen, 1988)).
Figure 5-10.Time evolution of the particle size dis-
tribution in Al–1 at.% Ag during aging at 413 K (af- ter Ernst and Haasen, 1988).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

380. ysis of composition fluctuations during
early-stage decomposition.
Analytical field ion microscopy (AFIM)
(Wagner, 1982; Miller et al., 1996) can fa-
vorably be applied for an analysis of ultra-
fine solute clusters in the early stages of
decomposition, as both the imaging resolu-
tion of the FIM (see Fig. 5-2) and the spatial
resolution of microanalysis of the integrated
time-of-flight spectrometer (‘atom probe’)
are sufficient, i.e., <1 nm and ª2 nm, re-
spectively. (Although the atom probe de-
tects single atoms from the probed volume,
the requirement for the analysis to be sta-
tistically significant confines the microana-
lytical spatial resolution to ≥2 nm.) As is
shown in Figs. 5-2a–c and 5-19c, both the
morphologies of small particles and their
three-dimensional arrangements can also
be determined in the FIM, at least for pre-
cipitates with sufficient contrast.
The volume sampled during an atom
probe FIM analysis is rather small (typi-
cally about 200 nm
3
). Hence, in order to
obtain statistically significant data con-
cerning the average size (R
¯
) and the num-
ber density (N
v) of the precipitates, the lat-
ter ought to exceed 10
23
m
–3
. Since after
the early stages of precipitation N
vde-
crease with time (see Sec. 5.2.1), AFIM
shows its full potential as a microanalytical
tool in studies of the early stages of precip-
itation, during which R
¯
is commonly small
andN
vsufficiently large (Haasen and Wag-
ner, 1985).
With respect to analyses of the spatial ar-
rangement and the composition of precipi-
tated phases, the versatility of the AFIM
was considerably improved by the devel-
opment of the tomographic atom probe
(TAP) (Blavette et al., 1993, 1998; Miller
et al., 1996). This instrument allows for
the three-dimensional reconstruction of a
small volume of the microscopically heter-
ogeneous material on a sub-nanometric
scale (Fig. 5-2d). The spatial distribution
of each chemical species can be directly
observed with a spatial resolution better
than 0.5 nm (Auger et al., 1995; Pareige et
al., 1999; Al-Kassab et al., 1997). The vol-
ume typically sampled is about 15×15
×100 nm
3
and, thus, about a hundred times
larger than the volume analyzed with the
AFIM.
5.3.1.2 Scattering Techniques
The time evolution of the structure of
supersaturated alloys, as well as of oxides
and polymer blends undergoing phase sep-
aration, can be analyzed by means of small
angle scattering of X-rays (SAXS), neu-
trons (SANS), and light. Light scattering is
of course confined to transparent speci-
mens, in which the domain sizes of the
evolving second phase must be of the order
of ≈1 μm. It has been successfully applied
in studies of decomposition, e.g., of a poly-
mer mixture of polyvinyl methyl ether 
(PVME) and polystyrene (PS) (Snyder and
Meakin 1983a, b, 1985), and of various
glass-forming oxide systems (Goldstein,
1965; Rindtone, 1975).
In principle, SAXS (Glatter, 1982) and
SANS (Kostorz, 1979; Sequeira et al.,
1995) provide access to a structural analy-
sis of unmixing alloys, in both the early
stages where the composition fluctuations
can be small in spatial extension and in am-
plitude, as well as in the later stages of de-
composition where R
¯
and N
vmay have at-
tained values which are quite unfavorable
for a quantitative analysis by TEM or FIM.
In addition, evaluation of the Laue scatter-
ing allows the remaining supersaturation of
solute atoms in the matrix to be deter-
mined. 
The metallurgist is left with the problem
of extracting in a quantitative manner all
the structural data contained in scattering
328 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
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381. 5.3 Experimental Techniques for Studying Decomposition Kinetics 329
curves such as are shown in Fig. 5-11. This
is often not trivial (Glatter, 1982), in partic-
ular for concentrated alloys (i.e., for most
technical two-phase alloys). Here the scat-
tering curve reveals a maximum (at posi-
tion 
k
mand with height S
m(k
m) in Fig.
5-11) which results from an interparticle
interference of the scattered waves (Kos-
torz, 1979). In general, the interparticle
interference function is not known, thus
impeding a straightforward quantitative
analysis. Moreover, the composition and
morphology of the fluctuations are re-
quired in order to perform such an analysis.
For many alloys this information can only
be obtained from AFIM or AEM. These
complicating factors often demand the
above-mentioned microanalytical tools to
be employed jointly, rendering experimen-
tal studies of decomposition rather difficult
and tedious. 
The small angle scattering intensity is
proportional to the structure function S(
k,
t); 
kis the scattering vector with k∫|k|=
4
psin (q/2)/l; lis the wavelength of the
X-rays or neutrons and 
qthe scattering an-
gle. For binary alloys S (
k, t) is the Fourier
transform of the two-point correlation
function at time t (Langer, 1975):
(5-16)
G(|r–r
0|,t)=·[c (r,t)–c
0][c(r
0,t)–c
0]Ò
The right-hand side in Eq. (5-16) denotes
the non-equilibrium average of the product
of the composition amplitudes at two dif-
ferent spatial positions, rand r
0, in the al-
loy with average composition c
0.
S(
k,t) contains all the structural infor-
mation on the phase-separating system.
Many theories and computer simulations
dealing with phase separation yield the ev-
olution of S(
k,t) and predict the shift of
k
m(which is a measure of the average
spacing of solute clusters) and the growth
of S
m(k
m) with time (Langer et al., 1975;
Marro et al., 1975, 1977; Binder et al.,
1978).
Hence, SAXS and SANS curves meas-
ured after different aging times can be di-
rectly compared with the predictions from
various theoretical kinetic concepts (see
Secs. 5.5.4, 5.5.6 and 5.8.1).
In ternary systems the situation is much
more complicated. For a substitutional ter-
nary alloy, there are three linearly indepen-
dent pair correlation functions with three
related partial structure functions (de Fon-
taine, 1971, 1973), which linearly combine
to the measured SAS intensity. An unam-
biguous characterization of the kinetics of
phase separation in ternary substitutional
alloys requires the three partial structure
functions to be determined separately. This
has been attempted by employing the
‘anomalous small angle X-ray scattering’
technique for investigations of phase separ-
ation in Al–Zn–Ag (Hoyt et al., 1987),
Cu–Ni–Fe (Lyon and Simon, 1987, 1988)
and Fe–Cr–Co (Simon and Lyon, 1989).
Unlike for X-rays, for thermal neutrons
the atomic nuclear scattering length is not
Figure 5-11. SANS curves of Cu-2.9 at.% Ti single
crystals aged for the given times at 350°C. Note that
even the homogenized and quenched (‘hom’) speci-
men yields SANS intensity, indicating that phase
separation has occurred during quenching (Eckerlebe
et al., 1986).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

382. monotonically dependent on the atomic
number (Kostorz, 1979). For this reason
use of SANS is more universal and is fre-
quently superior ro SAXS for decomposi-
tion studies of binary alloys with mainly
transition metal constituents having similar
atomic numbers. Consequently, SAXS has
only been used extensively for studies of
the unmixing kinetics in Al–Zn where the
difference (this, in fact, controls the con-
trast in SAS) in atomic scattering lengths
for X-rays is sufficiently large (Rundman
and Hilliard, 1967; Hennion et al., 1982;
Forouhi and de Fontaine, 1987).
Furthermore, if the two phases differ not
only in composition but also in magnetiza-
tion, both nuclear and magnetic SANS
curves can be recorded (see Fig. 5-12).
These two independently measurable
SANS curves sometimes even allow the
composition of the scattering centers to be
determined, e.g., for Fe–Cu, where dia-
magnetic copper-rich particles precipitate
in the ferromagnetic a -Fe matrix (Kamp-
mann and Wagner, 1986), or for phase-sep-
arated amorphous Fe
40Ni
40P
20(Gerling et
at., 1988).
Due to relatively simple calibration pro-
cedures in SANS experiments, the scat-
tered intensity which is expressed in terms
of the coherent (nuclear or magnetic) scat-
tering cross-section per unit volume,
d
S(k,t)/dW,can be measured in absolute
units. This is directly related to the struc-
ture function S(
k,t) via (Hennion et al.,
1982):
(5-17)
where b
–
Mand b
–
pare the locally averaged
(nuclear or magnetic) scattering lengths of
the matrix and of the solute-rich clusters,
respectively, and 
W
–
is the mean atomic vol-
ume.
5.3.2 Experimental Problems
5.3.2.1 Influence of Quenching Rate
on Kinetics
For studies on decomposition kinetics
the alloy is commonly homogenized in 
the single-phase region at T
H(Fig. 5-1),
quenched into brine and subsequently iso-
thermally aged at T
A. In order to capture
the initial stages of the decay of the super-
saturated solid solution, both the quench-
ing and the heating rate to T
Aare required
to be sufficiently high in order to avoid
phase separation prior to isothermal aging.
This can be comfortably achieved in alloys
with small supersaturations. It is, however,
often a problem (or even impossible) for
alloys with large supersaturations and/or
small interfacial energies, where the nucle-
ation barriers are small (see Secs. 5.2.3 and
5.5.1) and, hence, the nucleation rates
large.
The driving force for unmixing in Al-1
at.% Ag at 140°C is rather small
(≈0.5 kJ/mol). Nevertheless, because of the
extremely small interfacial energy of only
d
d
Mp
S
W W
(,) ( – ) (,)kkk ktbbSt=
1
2
2
330 5 Homogeneous Second-Phase Precipitation
Figure 5-12.Nuclear and magnetic SANS curves of
Fe–1.4 at.% Cu aged for 200 h at 400°C. From the
ratio of magnetic and nuclear scattering intensities,
the composition of the copper-rich clusters can be de-
rived. Theoretical curves are shown as full lines
(Kampmann and Wagner, 1986).www.iran-mavad.com 
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383. 5.3 Experimental Techniques for Studying Decomposition Kinetics 331
≈0.01 J/m
2
(Le Goues et al., 1984c), the
barrier to nucleation is so low that precipi-
tation commences instantaneously almost
as soon as the solvus line is crossed. Thus,
except for extremely high quenching rates,
the non-equilibrium single-phase state can-
not be frozen in and the as-quenched mi-
crostructure already contains a large num-
ber of small GP zones (Fig. 5-13).
The accessible quenching rates have
been found to be insufficient for a suppres-
sion of phase separation during quenching
of homogenized Cu–Ti alloys with Ti con-
centrations exceeding ≈2.5 at.%. This is
discernible from the SANS curve of the as-
quenched specimen (Fig. 5-11) (see Sec.
5.7.5).
Further problems result from the fact
that the decomposition kinetics at T
Aare
strongly influenced by the concentration of
quenched-in excess vacancies. This de-
pends strongly on the chosen homogeniza-
tion temperature and quenching rate.
Fig. 5-14 shows SANS curves from
specimens that were solution-treated at the
given homogenization temperature T
H.
Following quenching, each specimen was
aged for 10 min at 350°C. The maximum
in the SANS curves is found to be higher,
and its position lower, the higher the ho-
mogenization temperature. This indicates
that phase separation in the early stages
progresses the faster the higher the homog-
enization temperature is chosen. Obviously,
in Cu–Ti, the concentration of quenched-in
excess vacancies is correlated to T
Hand
significantly influences the decomposition
Figure 5-13.HREM micrograph of GP zones
formed during quenching of Al–1 at.% Ag. The
smallest GP zones have diameters of only ∂1nm
(reproduced by courtesy of P. Wilbrandt (Ernst et al.,
1987)).
Figure 5-14.SANS curves
of Cu–2.9 at.% Ti quenched from the given homogeniza- tion temperatures T
H, and
(apart from the bottom curve) subsequently aged for 10 min at 350°C (Ecker- lebe et al. 1986).www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

384. kinetics. This effect may be minimized if
the alloy is first solution-treated at high T
H
and subsequently equilibrated at a homog-
enization temperature slightly above the
solvus temperature prior to quenching.
The inconsistency in the exact mode and
kinetics of decomposition in Al–22 at.%
Zn, investigated by different authors
(Rundman and Hilliard, 1967; Gerold and
Merz, 1967), has been explained by the
presence of different quenching rates
which unavoidably lead to different states
of solute clustering in the as-quenched
specimens (Agarwal and Herman, 1973;
Bartel and Rundman, 1975).
The problems stated above make experi-
ments devised for examination of the vari-
ous spinodal theories particularly difficult.
5.3.2.2 Distinction of the Mode
of Decomposition
The criterion that must be satisfied in or-
der to distinguish s pinodal decomposition
(s.d.) from a nucleation and growth (n.g.)
reaction is to prove by any microanalytical
technique that the amplitude of the compo-
sition modulations of an alloy deeply
quenched into the miscibility gap increases
with time during the initial stages of phase
separation (see Sec. 5.2.2 and Fig. 5-6).
Even apart from the problem outlined in
the preceding section, this is a rather diffi-
cult task, since the cluster diameters or the
modulation wavelengths in most of the
more concentrated alloys investigated so
far have been found to range below the 
resolution limit of composition analysis,
e.g. ≈2 nm for AFIM. Figure 5-15 reveals
that the compositions of the solute clusters
in Ni–36 at.% Cu–9 at.% Al (Liu and 
Wagner, 1984) and in a hard-magnetic
Fe–29 at.% Cr–24 at.% Co (Zhu et al.,
1986) alloy have reached their equilibrium
values and, hence, remain constant once
the clusters have attained sizes that are ac-
cessible to chemical analysis by AFIM
(3 nm and 1.8 nm, respectively). (In fact
both ternary alloys may be regarded as
pseudobinary. Thus the thermodynamic
considerations of Sec. 5.2.2 still apply.)
The observed features suggest the reaction
in both alloys to be of the n.g. type. How-
ever, it cannot be ruled out that these clus-
332 5 Homogeneous Second-Phase Precipitation
Figure 5-15.a) Composition and diameter D
–
, of  g¢-
precipitates in Ni–36 at.% Cu–9 at.% Al as a func-
tion of aging time at 580°C. The corresponding pre-
cipitate microstructure is  shown in Fig. 5-2a–c (Liu
and Wagner, 1984). b) Composition and diameterr D
–
of the a
1(matrix) and a
2(precipitate ) phases in
Fe–29 at.% Cr–24 at.% Co as a function of aging
time at 640°C (Zhu et al., 1986).www.iran-mavad.com 
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385. 5.3 Experimental Techniques for Studying Decomposition Kinetics 333
ters result from a s.d. reaction which was
terminated after even shorter aging times
than could be covered in these studies.
Fig. 5-16 shows the interconnected pre-
cipitate microstructure of another hard-
magnetic alloy, Fe–29 at.% Cr–14 at.%
Co–21 at.% Al–0.15 at.% Zr, aged to yield
its optimum magnetic properties. In order
to characterize the decomposition process
with respect to the distribution of Cr
between the two phases, atom probe micro-
analysis was performed.
The measured Cr concentration of the
darkly imaging precipitating a
2-phase as a
function of aging time at T= 525 °C and 
T= 600°C is shown in Fig. 5-17a. During
aging at 600°C for less than 5 min it in-
creases continuously before it attains its
constant equilibrium value. This behavior
indicates unequivocally a spinodal decom-
position mechanism. At the lower aging
temperature of T
A= 525°C, the Cr ampli-
tude grows more slowly deeper within the
miscibility gap due to the slower diffusion,
but ultimately reaches higher equilibrium
concentrations in the a
2-precipitates, thus
showing the spinodal behavior even more
clearly. The observed increase in Cr con-
centration in the a
2-phase with decreasing
aging temperature is due to the widening of
the miscibility gap. As the Cr concentra-
tion in the a
2-phase increases, the propor-
tions of Co and Fe are reduced in this phase
and are correspondingly increased in the
a
1-phase. Atom probe analysis yields a
Fe/Co ratio of 3:1 for both the a
1- and a
2-
phases. This ratio corresponds to the tie-
line in the ternary system along which
Figure 5-16.Field ion image of Fe–29 Cr–14 Co–2
Al–0.15 Zr (at.%) aged into its optimum magnetic
state (Zhu et al., 1986). 
Figure 5-17.a) Evolution of the Cr concentration at
525°C and 600°C in the Cr-rich a
2-phase as a func-
tion of aging time. b) Composition of the Fe-rich (a
1)
and Cr-rich (a
2) phase after aging at 525°C for the
given times (in minutes). The dashed line corre- sponds to the tie-line with a Fe/Co ratio of 3/1 (Zhu et al., 1986).www.iran-mavad.com
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386. spinodal decomposition proceeds. Fig.
5-17b shows the compositions of the iron-
rich and chromium-rich phases for various
aging times at 525°C and the tie-line corre-
sponding to an Fe/Co ratio of 3:1.
This study is one of the very rare ones on
metallic alloys to have identified a decom-
position reaction unequivocally to be of the
spinodal type.
Frequently, the occurrence of either
quasi-periodically aligned precipitates
which give rise to side-bands in X-ray or
TEM diffraction patterns (Fig. 5-18) or of a
precipitated phase with a high degree of
interconnectivity (e.g., Fig. 5-16) has been
employed as a unique criterion by which to
define an alloy as spinodal. However, as
will be pointed out in Secs. 5.4.1 and 5.5.5,
morphology alone cannot be used to unam-
biguously distinguish spinodal decomposi-
tion from a nucleation and growth reaction
(Cahn and Charles, 1965).
5.4 Precipitate Morphologies
5.4.1 Experimental Results
There is a wide variety of different
shapes in individual precipitates and of
morphologies in precipitate microstruc-
tures. This is illustrated in Fig. 5-19 for
three different Cu–Ni-based alloys with
ternary additions of Al, Cr or Fe, the pre-
cipitated volume fraction (f
p) of which is
rather large, e.g., f
p(CuNiAl) ≈0.20, f
p
(CuNiCr) ≈0.19, and f
p(CuNiFe) ≈0.37.
Aging of Ni–37 at.% Cu–8 at.% Al for
167 h at 580 °C yields randomly distrib-
uted spherical particles with R
–
≈11 nm and
a number density of ≈3.6¥10
22
m
–3
(Fig.
5-19a). In contrast, TEM of Cu–36 at.%
Ni–4 at.% Cr aged for 240 h at 650 °C re-
veals a modulated precipitate structure
with cuboidal particles aligned along the
334 5 Homogeneous Second-Phase Precipitation
Figure 5-18.TEM micrographs of Cu–48 at.% Ni–8
at.% Fe aged for a) 8 h, b) 23 h, and c) 65 h at 500°C.
Each insert shows two satellites around the bright
(002) matrix reflections. The distance between the
satellites and the fundamental reflection is inversely
proportional to the wavelength of composition mod-
ulations or to the precipitate spacing in modulated
structures (reproduced by courtesy of R. P. Wahi 
(Wahi and Stajer, 1984)).www.iran-mavad.com 
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387. 5.4 Precipitate Morphologies 335
three ·100Òdirections (Fig. 5-19 b). Re-
placing Al or Cr by Fe leads to a mottled 
(‘sponge-like’) precipitate microstructure
in Cu–48 at.% Ni–8 at.% Fe aged for 8 h at
500 °C. The three-dimensional intercon-
nectivity of the mottled structure becomes
discernible by a reconstruction of a se-
quence of FIM images which were taken at
various distances underneath the original
surface of the FIM specimen (Fig. 5-19 c).
Modulated structures have been pre-
dicted (Cahn, 1965; Hilliard, 1970) to
evolve from spinodal decomposition in
elastically anisotropic cubic matrices, ow-
ing to the tendency to minimize the cohe-
rency strain energy (see Sec. 5.2.4, Eq. 
(5-12)), Furthermore, in material which is
either isotropic or for which the elastic en-
ergy (see Eq. (5-12)) is negligibly small,
spinodal decomposition is predicted to
generate an interconnected mottled precip-
itate morphology. Computer simulations of
the latter case, which were based on a
superposition of randomly oriented sinu-
soidal composition fluctuations of fixed
wavelength, random phase shifts and a
Gaussian distribution of amplitudes,
Figure 5-19.a) TEM dark field image of  g¢-precipi-
tates in Ni–37 at.% Cu–8 at.% Al aged for 167 h at
580°C (Wagner et al. 1988). b) TEM bright field im-
age of Cu–36 at.% Ni–4 at.% Cr aged for 240 h at
650°C displaying particle alignment along the ·100Ò
matrix directions (Wagner et al., 1988). c) FIM mi-
crographs of Cu–48 at.% Ni–8 at.% Fe aged for 8 h
at 500°C. Between each FIM micrograph 2 nm of the
specimen surface were removed (‘field evaporated’
(Wagner, 1982)) in order to reveal the three-dimen-
sional arrangement of the brightly imaged (Ni, Fe)-
rich precipitated phase. In a
the precipitates C, D, E,
F, G are apparently isolated; after having removed 2 nm, they have merged into one large extended par- ticle (b
). At still greater depth (d; 6 nm) ‘particle’ C
appears again to be isolated from the others by the darkly imaged matrix (reproduced by courtesy of W. Wagner (Piller et al., 1984)).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

388. yielded an interconnectivity of the two
conjugate phases for volume fractions
ranging from ≈0.15 to  ≈0.85 (Cahn, 1965).
The simulated morphological pattern (Fig.
5-20) resembles the FIM images shown in
Figs. 5-16 and 5-19c remarkably well.
This has made it quite tempting to define
these alloys and, more generally, all alloys
displaying interconnected or quasi-periodi-
cal morphologies, as spinodal alloys. In
general, however, in more concentrated al-
loys where the number density of clusters
of the new phase is large and, hence, the
intercluster spacing small, interconnectiv-
ity and quasi-periodicity in the late-stage
microstructure may result from other
mechanisms. Examples of such mecha-
nisms are coalescence of neighboring parti-
cles (e.g., in the BaO–SiO
2system) or se-
lective coarsening of elastically favorably
oriented particles (e.g., in Ni–Al (Ardell
and Nicholson, 1966; Doi et al., 1988)).
Hence, in essence, the distinction between
a spinodal reaction and nucleation and
growth cannotbe based solely on any spe-
cific morphological features but requires a
complete study of the evolution of the new
phase with aging time (see Sec. 5.5.5).
5.4.2 Factors Controlling the Shapes
and Morphologies of Precipitates
In the case of homogeneous nucleation,
the precipitating particles are commonly
coherent with the matrix. Their shape is
controlled by the rather complex interplay
of various factors, such as the magnitude
and anisotropy of the interfacial energy, the
difference in the elastic constants between
matrix and precipitate, and the crystal
structure of the latter (Khachaturyan,
1983).
The complexity of the situation is illus-
trated in Fig. 5-21a (Lee et al., 1977).
There the anisotropic elastic strain energy
DF
el/d
2
of an ellipsoidal Ag-rich particle
with different orientational relationships
with respect to a Cu or Al matrix is plotted
as a function of the aspect ratio K∫T/R,
where Tand Rdenote the two axes of an el-
lipsoid of revolution.
Assuming the interfacial energy to be
isotropic, Fig. 5-21a reveals that the mini-
mum strain energy is obtained for platelet-
shaped particles (K∂1), the cubic direc-
tions of which lie parallel to those of the Al
or Cu matrices.
More generally, theory predicts (Lee et
al., 1977) platelets to have the minimum
and spheres to have the maximum strain
energy if the precipitated phase is elasti-
cally softer than the matrix, regardless of
the orientation relationship and elastic an-
isotropy. For the reverse situation of a hard
particle embedded in a softer matrix, the
sphere represents the minimum strain en-
ergy shape.
However, the elastic constants of the
precipitating phase are often not available,
rendering the computation of the shape
with minimum strain energy difficult. In
336 5 Homogeneous Second-Phase Precipitation
Figure 5-20.Computer-simulated cross-section of a
spinodal structure in an isotropic solid displaying the
interconnectivity of the two incipient phases 1 and 2
with equal volume fraction. (After Cahn, 1965).www.iran-mavad.com 
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389. 5.4 Precipitate Morphologies 337
particular, this holds true if the nucleating
phase is a metastable transition phase. 
In essence, however, the adopted shape
is determined by the balance between the
interfacial Helmholtz energy D F
a/band the
elastic Helmholtz energy D F
el. DF
a/bis
minimized for particles with spherical or
faceted shape (e.g., Figs. 5-9 and 5-19)
whereas DF
el, which is related to the parti-
cle volume, commonly prefers a platelet-
like morphology (e.g., Fig. 5-21a). Thus,
during the early stages of precipitation,
DF
a/bis the dominating term, whereas
DF
elprevails in the limit of large particles. 
The nature of this elastic stress-induced
precipitate–shape transition has been in-
vestigated by Johnson and Cahn (1984) us-
ing bifurcation theory. Assuming an elasti-
cally isotropic inhomogeneous system and
by restraining the morphology of the parti-
cle to be ellipsoidal it was possible to ad-
dress analytically the structure of the shape
bifurcation. For example, in two dimen-
sions they found that at a critical size the
shape of a circular precipitate will change
to an ellipse in a smooth manner as the size
of the precipitate is increased; the shape bi-
furcation is supercritical. A supercritical
bifurcation is analogous to the change in
the order parameter on passing through a
second-order phase transition. In contrast,
in three dimensions, transcritical bifurca-
tions (first-order transitions) are possible.
Such a bifurcation implies that discontinu-
ous changes in particle shape will occur as
the particle increases its size.
The work of Johnson and Cahn clearly
showed the importance of particle shape
bifurcations in understanding microstruc-
tural evolution in systems with misfitting
particles. Their study, however, was re-
stricted to ellipsoidal particle shapes. This
was remedied in the work of Thompson and
Voorhees (1999) where the equilibrium
shape of a particle in an elastically aniso-
tropic homogeneous system with isotropic
Figure 5-21.a) Strain energy vs. aspect ratio of an Ag precipitate in Cu and Al matrices. The numbers 1, 4, 5
and 7 designate different orientation relationships between matrix and precipitate. The lowest strain energy is
obtained if the cubic directions of both particle and matrix are parallel to each other (curves 1), 
dis the linear
misfit. After Lee et al. (1977). b) The variation in a parameter, a
R
2
, as a function of L. a
R
2
is used to characterize
the symmetry of a given family of equilibrium shapes: it is zero if the shape is four-fold symmetric and non-zero
if it is two-fold symmetric (Thompson et al., 1994).
a) b)www.iran-mavad.com
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390. interfacial energy was determined numeri-
cally, see Fig. 5-21b. Here the equilibrium
shapes were found as a function of the di-
mensionless parameter L=
d
2
C
44l/s
ab
where C
44is an elastic constant in a system
with cubic elastic anisotropy, and lis the
size of a particle, e.g., l =3V
1/3
/4pwhere
Vis the volume of a precipitate, Lis the ra-
tio of the characteristic elastic to interfacial
energy. When the particle is small, interfa-
cial energy is the dominant factor in setting
the particle shape and as L increases, the
effects of elastic stress become important.
When Lis small and the equilibrium shape
has a fourfold symmetry, a cube-like mor-
phology with rounded corners and rounded
sides will result. As Lincreases, at a criti-
cal size there is a supercritical bifurcation
to one of two plate-like shapes oriented
along the elastically soft directions. How-
ever, due to the presence of interfacial en-
ergy, the equilibrium shapes above the bi-
furcation point are not plates for any finite
L. In the limit of infinite Lthe equilibrium
shape is an infinitely long plate of zero
thickness (Khachaturyan, 1983). The loca-
tion of the bifurcation point is a function of
both the elastic anisotropy and the differ-
ence in elastic constants between the
phases (Schmidt and Gross, 1997). Three-
dimensional calculations have also been
performed where the particles are not con-
strained to be of a simple geometric shape
(Mueller and Gross, 1998, Lee, 1997;
Thompson and Voorhees, 1999).
Another qualitative difference between
the possible equilibrium shapes of a parti-
cle in the presence of stress and that in the
absence of stress is that the morphology of
misfitting particles can be non-convex. In
the absence of stress, regardless of the type
of interfacial energy anisotropy, the equi-
librium shape must be convex. In contrast,
Thompson and Voorhees (1999) found
metastable non-convex shapes in an elasti-
cally anisotropic system with a tetragonal
misfit, and Schmidt and Gross (1997)
found strongly non-convex global energy
minimizing, pincushion-like shapes when
the particle is sufficiently soft in an elasti-
cally anisotropic system. Similar pincush-
ion-like precipitates have been observed
experimentally, but in a system with a
much smaller difference in elastic con-
stants (Maheshwari and Ardell, 1992). Em-
phasizing the difficulty in obtaining true
equilibrium shapes experimentally, Wang
and Khachaturyan (1995) have suggested,
however, that the shapes observed by Ma-
heshwari and Ardell are kinetic in origin.
In an internally nitrided Fe 3 at.% Mo al-
loy, (Fe, Mo)
16N
2-type precipitates were
found to nucleate as thin platelets with 
K≈0.1 and with an undistored {100}
a-Fe
plane common to both the a-matrix and the
nitrides. The observed morphology, as well
as the {100}
a-Fehabit planes, have been
accurately predicted by applying macro-
scopic linear elastic theory to tetragonal
Fe
16N
2precipitates in ferritic iron, and as-
suming the interfacial energy to be iso-
tropic (Khachaturyan, 1983). The latter as-
sumption, however, is debatable, since dur-
ing aging at 600°C (Fe, Mo)
16N
2platelets
only grow markedly in diameter; their
thickness remains almost constant (Fig.
5-22). This has been interpreted in terms of
the large difference between the interfacial
energy of the habit plane (≈ 0.05 J/m
2
) and
that of the peripheral plane (≈0.3 J/m
2
)
(Wagner and Brenner, 1978).
Under certain conditions, the elastic or
magnetic interaction of precipitates with
external stress fields or magnetic fields al-
lows for the generation of anisotropic,
highly oriented precipitate microstructures
which are sometimes of technological im-
portance. If the transformation strain is
non-spherical, such as for Fe
16N
2in ferritic
Fe–N, the particles may interact with an
338 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

391. 5.5 Early Stage Decomposition Kinetics 339
externally applied stress field. The result-
ing elastic interaction energy depends on
the particular orientation of the particle
with respect to this field. During aging, this
causes a selective coarsening of favorably
oriented particles at the expense of the en-
ergetically less favorably oriented ones.
For instance, if the originally isotropically
distributed Fe
16N
2platelets undergo coars-
ening in the presence of an external tensile
stress applied along [001], a highly
oriented (anisotropic) precipitate structure
will result (Ferguson and Jack, 1984,
1985), with the habit planes of all particles
being parallel to each other and perpendic-
ular to [001]. In addition, these platelets
show a strong uniaxial magnetic anisotropy
with the direction of easy magnetization
parallel to the plate normal. This magnetic
property can also be used for completely
orienting the platelets during aging in an
external magnetic field (Sauthoff and
Pitsch, 1987).
5.5 Early Stage Decomposition
Kinetics
As outlined in Sec. 5.2.3, the early
stages of unmixing of a solid solution
quenched into the miscibility gap are trig-
gered by the growth and the decay of con-
centration fluctuations. Basically, the ob-
jective of any theory dealing with the ki-
netics of early stage decomposition is the
prediction of the particular shape, ampli-
Figure 5-22.FIM micrographs of (Fe, Mo)
16N
2
platelets (bright) in an a-iron matrix (dark). During
coarsening at 600°C for the given times, the aspect
ratio decreases from ∫0.1 (t = 0) to ∫ 0.04 (623 h),
whereas the platelet thickness increases only from
∫0.7 nm to ∫ 1.0 nm. The platelets intersect the sur-
face of the semi-spherical field ion tip and thus ap-
pear to be curved (Wagner and Brenner, 1978).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

392. tude and spatial extension (or number of at-
oms) of a solute fluctuation which becomes
critical and, hence, stable against decay.
Once formed, the critical fluctuations ren-
der the supersaturated alloy unstable with
respect to further unmixing.
According to Sec. 5.2.4 and referring to
Fig. 5-6a, nucleation theories consider the
formation rate of stable nuclei, the latter
representing spatially localized solute-rich
clusters (‘particles’ or ‘droplets’) with
large concentration amplitudes. In this con-
text, a distinction is made between classi-
cal (Sec. 5.5.1.1) and non-classical nuclea-
tion theory (Sec. 5.5.1.4) depending on
how the Helmholtz energy of the non-uni-
form solid solution containing the cluster
distribution is evaluated. Once the Helm-
holtz energy has been specified, the equi-
librium distribution of heterophase fluctua-
tions and the nucleation barrier (Sec. 5.2.3)
can be calculated. The nucleation rate is
then obtained as the transfer rate at which
smaller clusters attain the critical size.
A ‘composition wave’ picture rather
than a ‘discrete droplet’ formalism is em-
ployed in the spinodal theories (Sec. 5.5.4)
which, as another approach, describe the
early-stage decomposition kinetics in
terms of the time evolution of the ampli-
tude and the wavelength of certain stable 
‘homophase’ fluctuations (Fig. 5-6b).
There are several papers dealing with the
early-stage unmixing kinetics. Russel
(1980) and Aaronson and Russel (1982)
consider both classical and non-classical
nucleation phenomena from a more metall-
urgical point of view; the formulation of a
microscopic cluster theory of nucleation
can be found in the article by Binder and
Stauffer (1976). The comprehensive arti-
cles by Martin (1978) and by Gunton and
Droz (1984) disclose developments of both
nucleation and spinodal concepts not only
to the theoretican but also to the materials
scientist. For more recent developments,
see the chapter by Binder and Fratzl (2001).
5.5.1 Cluster-Kinetics Approach
5.5.1.1 Classical Nucleation –
Sharp Interface Model
Suppose the homogenized solid solution
is quenched not too deeply into the meta-
stable regime of the miscibility gap (e.g., to
point 1 in Fig. 5-5a). There it is isother-
mally aged at a temperature sufficiently
high for solute diffusion. After a certain
time, a distribution of microclusters con-
taining iatoms (i -mers) will form in the
matrix.
Generally, classical nucleation is now
based on both a static and a dynamic part.
In the static part the changes of Helmholtz
energy associated with the formation of an
i-mer and the cluster distributionf(i) must
be evaluated. In the dynamic part the kinet-
ics of the decay of the solid solution which
now is described by the given distribution
of non-interacting microclusters, are calcu-
lated in terms of the time evolution off(i);
ultimately this will furnish the formation
rate of stable clusters, i.e. the nucleation
rate.
Classical nucleation theory treats the so-
lute fluctuations as droplets which were cut
from the equilibrium precipitate phase b
and embedded into the amatrix; in this
capillarity or droplet model the interface
between aand bis assumed to be sharp,
e.g., Fig. 5-6a. In essence, the approxima-
tion reduces the number of independent
variables which characterize a cluster, and
which may vary during the nucleation pro-
cess (e.g., the solute concentration, the
atomic distribution within the cluster, its
shape, the composition profile across the
interface, etc.) and, hence, determine the
Helmholtz energy of the system, to virtu-
ally a single one. That is the number iof
340 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

393. 5.5 Early Stage Decomposition Kinetics 341
atoms contained in the cluster or, in terms
of a more macroscopic picture, the radius R
of the droplet; (4
p/3)R
3
=iW
bif W
bde-
notes the atomic volume in the droplet.
Once the small nucleus containing only a
few atoms is treated as a droplet of the new
phase bhaving bulk properties (which, in
fact, is a rather debatable approximation),
the interfacial Helmholtz energy and the
Helmholtz energy are also considered to be
entirely macroscopic in nature.
The formation of a coherent droplet with
radius Rwhich gives rise to some elastic
coherency strains, leads to a change of
Helmholtz energy,
(5-18)
where, according to Sec. 5.2.4, (Df
ch+ 
Df
el) is the driving force per unit volume
and 
s
abthe specific interfacial energy. 
The first term in Eq. (5-18) which scales
with R
3
accounts for the gain of Helmholtz
energy on forming the droplet (i.e., it is
negative). The second term which scales
with R
2
has to be expanded on forming the
interphase boundary and hence is a positive
contribution to DF(R).
Fig. 5-23 shows the dependence of the
two contributions in Eq. (5-18) on the
droplet radius. The resulting droplet forma-
tion energy DF(R) passes through a maxi-
mum at R ∫R* or i∫i* with 
(5-19)
Accordingly, clusters with R =R* are in
unstable equilibrium with the solid solu-
tion, i.e., the Helmholtz energy of the
system is lower if it contains clusters with
sizes below or beyond R* or i*. Therefore,
only clusters with radii that exceed the ra-
R
ff
i
ff
*
–( )
*
–( )
=
+
=
+
⎡
⎣
⎢
⎤
⎦
⎥
2
4
3
2
3
s
p s
ab
b
ab
DD
DD
chem el
chem el
or
W
DDD
ab
FR f f R R() ( )=+ +
ch el
4
3
4
32p
ps
dius R* of the critical nucleus are predicted
to grow continuously. This requires a fluc-
tuation to become a critical nucleus, first to
overcome the activation barrier for nuclea-
tion, or the nucleation energy,
(5-20)
As outlined in Sec. 5.2.4, Df
chemde-
creases strongly with decreasing supersatu-
ration, or, for a given concentration, with
increasing aging temperature (Fig. 5-1).
Correspondingly, since 
s
aband Df
elare
only weakly dependent on temperature,
both R* and D F* increase strongly with
decreasing supersaturation. Hence, the
droplet model ought to approximate the en-
ergetics of an unmixing alloy better the
smaller the supersaturation.
Once we have specified the free energy
of formation of a droplet in terms of its size
Ror i(Eq. 5-18)), the Helmholtz energy of
the system containing a number of f(i,t) of
non-interacting clusters of size iper unit
volume at time tis given as (Frenkel, 1939;
DD
DD
ab
3
FR F
ff
(*) *
()
≡=
+
16
3
2
p s
chem el
Figure 5-23.Schematic representation of Helmholtz
energy changes associated with cluster formation as
a function of cluster radius (R) or number iof atoms
in the cluster. DF
0
is the Helmholtz energy of the 
homogeneous solid solution, Zthe Zeldovich factor.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

394. Russell, 1980):
(5-21)
The entropy of mixing S
mixarises from
distributing the clusters on the available N
0
lattice sites per unit volume of the crystal.
The kinetics of early-stage unmixing are
governed by the change in the cluster size
distribution function f(i,t) with aging
time. Microscopically, this may occur via
different processes.
Volmer and Weber (1926), Becker and
Döring (1935) and Zeldovich (1943), the
works which most nucleation theories are
based on, assume the transitions between
size classes in an assembly of non-interact-
ing droplets to occur via the condensation
or evaporation of single solute atoms.
Hence, since only a transition between size
classes iand i+ 1 is allowed, its flux, J
iis
given as (Russell, 1980):
J
i=b(i)f(i,t) – a(i+1)f(i+1,t) (5-22)
where 
b(i) is the condensation rate and
a(i+ 1) the evoporation rate of a single
atom to a cluster of size ior from a cluster
of size (i + 1), respectively. 
In equilibrium, the fluxes J
imust vanish
and f(i,t) becomes identical with the equi-
librium cluster size distribution C(i) for
which F in Eq. (5-21) attains a minimum:
C(i)=N
0exp [–DF(i)/kT] (5-23)
It is worth noting that in this case the
number of critical nuclei C*∫C(i∫i*) is
proportional to exp (–DF */kT). Then, ac-
cording to Eq. (5-22), the condensation
rate and evaporation rate are related to each
other via
(5-24)
In the conventional nucleation theory it
is now assumed that the evaporation rate
ab() ()
()
()
ii
Ci
Ci
+=
+
1
1
FFifitTS=∑D( (
1
n
mix
),)–
a(i+ 1) derived for the equilibrium situa-
tion (‘principle of detailed balance’) is still valid for the non-equilibrated system where f(i,t) ≠C(i) and J
i≠0. Under such
an assumption, which becomes reasonable when i-mers are able to relax internally
between atomic condensation or evapora- tion, the flux of clusters between size classes iand i+ 1 is obtained from Eqs. 
(5-22) and (5-23) (Russell, 1980):
(5-25)
In the earliest theory on nucleation 
(Volmer and W eber; VW, 1926) it is as-
sumed thatf(i,t) ∫0 for clusters with
i>i* (or R>R*), and that clusters with R
>R* decay artificially into monomers,
thus keeping the matrix supersaturation about constant. The resulting quasi-steady- state distribution of cluster sizes is ob- tained from Eq. (5-23) and shown in Fig. 5-24). In this theory the steady-state nucle-
ation rate J
s
V–W
is obtained as the product
of the number C* of critical nuclei and the
rate 
bat which a single solute atom im-
pinges on the  critical nucleus rendering it supercritical, i.e.:
J
s
VW
=bC* =bN
0exp (–D F*/kT) (5-26)
One of the shortcomings of the VW the-
ory is that supercritical droplets with 
i>i* are assumed notto belong to the
cluster size distribution. This was cleared
up by the theory of Becker and Döring
(1935). In their theory (BD) the non-equi-
librium steady-state distribution of small
clusters with i∫i* is identical with that of
VW but unlike in the latter theory, clusters
with i*≤i≤i
c(i
cis a somewhat arbitrarily
chosen cut-off size) are considered to be-
long to the size distribution (Fig. 5-24).
JiCi
fi t
Ci
fit
Ci
iCi
fit Ci
i
i=
+
+
⎡
⎣
⎢
⎤
⎦
⎥
≈
∂
∂
⎡
⎣
⎢
⎤
⎦
⎥b
b() ()
(,)
()
–
(,)
()
–()()
(,)/ ()
1
1
342 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

395. 5.5 Early Stage Decomposition Kinetics 343
Hence, according to Eq. (5-22) a decay
or dissolution of supercritical droplets with
i>i* becomes a likely process in the BD
theory and is accounted for by the Zeldo-
vich factor Z. With the assumption that the
rate 
b∫b* at which a solute atom impinges
on a critical droplet is proportional to its
surface area, integration of the cluster flux
equation (Eq. (5-25)) yields the steady-
state nucleation rate J
S
BD
of the BD theory
as (Russell, 1980):
J
s
BD
=Zb* N
0exp (–D F*/kT) (5-27)
with
(5-28)
According to Eq. (5-27), nucleation is a
thermally activated process with an activa-
tion energy identical to that (DF*) of
forming a critical nucleus of size i* or R*.
Furthermore, like the thermodynamic
model that yields the number of critical 
nuclei proportional to exp (–DF*/kT) (Eq.
(5-23)), the kinetic treatment predicts the
steady-state nucleation rate also to be pro-
portional to the same exponential factor.
Z
kT
F
i
ii
=
∂
∂
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
–
*
/
1
2
2
2
12
p
∆
The Zeldovich factor Zis of the order of
1/20 to 1/40; graphically, its reciprocal
value corresponds approximately to the
width of the potential barrier, D F(R) or
DF(i), at a distance kTbelow the maxi-
mum (Fig. 5-23).
5.5.1.2 Time-Dependent Nucleation
Rate
It must be pointed out that the above
given steady-state nucleation theories do
not provide any information on the mo-
mentary cluster size distribution or on the
nucleation rate prior to reaching steady-
state conditions, i.e., the time-dependent
nucleation rate J*(t). Commonly the latter
is considered in terms of the steady-state
nucleation rate J
s
and an incubation period
tvia:
J*(t)=J
s
exp (–t/t) (5-29)
Steady-state will be achieved once the
clusters have attained sizes for which the
probability of redissolution is negligibly
small. Referring to Fig. 5-23 and recalling
the physical meaning of the Zeldovich fac-
tor Z, this will be the case for clusters with
i>i* +1/2Z(Feder et al., 1966). As the
gradient of DF(i) within the region 1/Zis
rather small, the clusters will move across
this region predominantly by random walk
with the jump frequency 
b*. The time
(5-30)
to cover the distance 1/Zby random walk is
identified with the incubation period.
So far it has been assumed that during
nucleation the supersaturation and, hence,
the driving force remains unchanged. At its
best this may be valid for extremely low
nucleation rates: in this case DF*, Zand 
t
may be considered time-independent quan-
tities. Russell (1980) proposed the nuclea-
t
b=
1
2*
Z
Figure 5-24.Quasi-stationary cluster size distribu-
tions of the Volmer–Weber (V–W) and Becker–
Döring (B–D) theory.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

396. tion process to terminate itself under
steady-state conditions if the incubation
time 
tis shorter than some critical time t
c.
If, however, 
t> t
c, phase separation will be
completed without steady-state ever having
been achieved. This situation, which in fact
is met in most decomposing alloys studied
up to now (see Sec. 5.7.4.1), is referred to
as catastrophic nucleation; under such
conditions DF*, Zand 
tbecome time de-
pendent. Assuming that nucleation ceases
once the diffusion fields and around pre-
cipitates with radius Rand composition c
b
e
embedded in a matrix with initial composi-
tion c
0begin to overlap (Fig. 5-6 a), and
furthermore that nucleation becomes un-
likely within a region R/
e(e≈1/10) around
the particle where the solute content has
decreased below (1 –
e) c
0, the critical time
t
cis estimated to be:
(5-31)
where D is the solute diffusion coefficient.
For a quantitative assessment of classi-
cal nucleation theory, the atomic impinge-
ment rate 
b* in Eq. (5-27) must be known.
For spherical nuclei it was evaluated to be
(Russell, 1970):
(5-32)
where ais the lattice parameter. 
b* is pro-
portional to the nucleus surface, as was as-
sumed in the original Becker–Döring the-
ory.
According to Sec. 5.4.2 the nucleus
shape with minimum energy may deviate
from the spherical one due to different en-
ergies of the interfaces (some may be co-
herent, some semi- or incoherent, Sec.
5.2.1) bounding the nucleus in different
crystallographic directions. It has been
shown (Chan et al., 1978) that in this case
b
p*
*
=
4
2
0
4
RDc
a
t
c
c DJ
c
e
s 2=
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥e
6
0
3
3
15 1
2
1β
()
/
the expressions for DF*, Z,  b* and tgiven
above for spherical clusters, have merely to
be multiplied with numerical factors but
otherwise remain unchanged. The magni-
tude of these numerical factors depends on
the particular equilibrium shape of the nu-
cleus. Ignoring volume strain energy, the
equilibrium shape of the nucleus can be de-
termined by means of the Wulff construc-
tion on a three-dimensional 
s
ab-plot (Her-
ring, 1953). The latter is obtained from a
radial plot of different vectors in every di-
rection e.g., vv
1, vv
2, vv
3in Fig. 5-25), the
length of which is proportional to the en-
ergy of the interface perpendicular to the
particular vector. The surface connecting
the tips of all vectors represents the polar
s
ab-plot, wherein the cusps indicate inter-
faces with good atomic matching , i.e., low
interfacial energies. Subsequently planes
344 5 Homogeneous Second-Phase Precipitation
Figure 5-25.Polar  s
ab-plot for particles with aniso-
tropic interfacial energies 
s
ab. The normals drawn at
the tip of each vector vvare the Wulff planes, the inner
envelope of which gives the equilibrium shape of the
precipitate. If the 
s
ab-plot displays deep cusps (e.g.,
for vv
1, vv
3, and vv
5) the Wulff construction yields a fac-
eted polyhedron.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

397. 5.5 Early Stage Decomposition Kinetics 345
are drawn (the so-called Wulff planes) per-
pendicular to the vectors vv
1, vv
2, vv
3, etc.
which intersect the 
s
ab-plot (Fig. 5-25).
The inner envelope of the Wulff planes
yields the equilibrium shape of the nucleus
with facets at the cusps of the 
s
ab-plot
(Martin and Doherty, 1976).
LeGoues et al. (1982) employed a near-
est-neighbor interaction, regular solution
model for a computation of the  
s
ab-plot as
a function of T/T
c(T
cis the critical temper-
ature) within the miscibility gap of an f.c.c.
solid solution. At low temperatures they
found the nucleus shape to be fully faceted
by {100} and {111} planes whereas at 0.5
T
cand near-zero supersaturation, the nu-
cleus shape can be rather well approxi-
mated by a sphere. They furthermore
showed that beyond T ≈0.4T
cthe parame-
ters DF*, Z, 
t, and b* which enter the time-
dependent (classical) nucleation rate (Eqs.
(5-27) and (5-29)) need no longer be cor-
rected for deviations from spherical shape.
Recently the question of to what extent
the energy of the coherent interphase boun-
dary depends on its crystallographic orien-
tation and on temperature has been read-
dressed by employing the cluster variation
method (CVM) with effective cluster inter-
action parameters (Sluiter and Kawazoe,
1996; Asta, 1996). The latter are specific
for the particular two-phase alloy system
under consideration. They can be derived
from the results of ab initiototal energy
calculations as was demonstrated by com-
puting the thermodynamic properties of
interphase boundaries between disordered
f.c.c. matrices and ordered (L1
2type) pre-
cipitates in Al–Li (Sluiter and Kawazoe,
1996) and Al–Sc (Asta et al., 1998) as well
as disordered GP zones (cf. Fig. 5-13) in
Al–Ag alloys (Asta and Hoyt, 1999).
Between T
cand T= 0.5 T
cthe CVM-calcu-
lated energies for the cube {100} and octa-
hedral {111} orientations of the flat inter-
face between Al-rich matrix and Ag-rich
GP zones are found to be equal within 5%,
and to increase from zero to about
30 mJ/m
2
if the temperature is lowered
from T
cto ~ 0.5 T
c. This result is compat-
ible with the spherical particle morphology
resolved in the HREM micrographs (Figs.
5-9 and 5-13).
5.5.1.3 Experimental Assessment of
Classical Nucleation Theory
A quantitative assessment of classical
nucleation theory in solids is inherently
difficult and thus has prompted only a few
studies. First of all, the range of supercool-
ing (or supersaturation) has to be chosen
such that nucleation is homogeneous and
the nucleation rates are neither unmeasure-
ably slow nor beyond the limits of catas-
trophic nucleation. Secondly, the quench-
ing rate must be sufficiently high in order
to avoid phase separation during the
quench but also sufficiently low in order to
avoid excess vacancies to be quenched in.
Servi and Turnbull (1966) studied in a
well-designed experiment homogeneous
nucleation kinetics of coherent Co-rich
precipitates in Cu–1 … 2.7 wt.% Co al-
loys. By using electrical resistivity meas-
urements, they could determine rather ac-
curately the precipitated volume fraction.
Assuming that the growth is diffusion-con-
trolled (Sec. 5.5.2), from the latter the par-
ticle density at the end of the precipitation
process could be derived as a function of
aging temperature and composition. The
thus indirectly obtained number density of
Co-rich particles, which was later corrobo-
rated by CTEM studies, agreed within one
order of magnitude with the value pre-
dicted by classical nucleation theory if the
interfacial energy was taken as ≈0.19 J/m
2
;
this value was derived from discrete lattice
calculations (Shiflet et al., 1981).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

398. The validity of classical nucleation the-
ory, as proven by the Servi–Turnbull study
on Cu–Co alloys, was later challenged by
LeGoues and Aaronson (1984). They
argued that the supersaturation employed
in the Servi–Turnbull investigations was
probably too high to avoid phase separa-
tion during quenching and also probably
too high to avoid concomitant coarsening
during the precipitation reaction. Employ-
ing a discrete lattice point model, which in-
corporates coherency strain energy, Le-
Goues and Aaronson first evaluated the 
‘window’ of temperatures (D T≈50°C) and
compositions (0.5 to 1.0 at.% Co) at which
homogeneous nucleation kinetics would be
neither too sluggish nor too fast, and at
which no interference with coarsening
would be expected. Prior to CTEM analy-
ses, the isothermally nucleated particles
had to be subjected to diffusion-controlled
growth in order to increase their radius be-
yond a certain size (R≈5 nm) at which the
Co-rich particles became easily discernible
in the CTEM. The experimental results
were interpreted in terms of classical nu-
cleation theory (Eq. (5-29)). The agree-
ment between the experimentally obtained
nucleation rates and the theoretically pre-
dicted ones was again rather good, thus
providing further support for the validity of
classical nucleation theory. Furthermore,
as is shown in Fig. 5-26, for smaller super-
saturations the nucleation energies DF*
and the critical radii R* as evaluated from
classical nucleation theory are almost iden-
tical to the corresponding quantities calcu-
lated from either the non-classical nuclea-
tion theory (cf. Sec. 5.5.1.4, Eq. (5-37)) or a
discrete lattice point theory. Hence, at least
for smaller supersaturation, the classical
theory predicts the nucleation rates about as
well as the two more sophisticated theories.
In another attempt to assess classical nu-
cleation theory, Kirkwood and coworkers
(Kirkwood, 1970; West and Kirkwood,
1976; Hirata and Kirkwood, 1977) studied
early-stage precipitation of 
g¢-Ni
3Al in
Ni–Al alloys, also using CTEM. They
found the 
g¢-Ni
3Al precipitate number den-
sity to decrease instantaneously upon ag-
ing, which is indicative of an extremely
fast nucleation process and the observation
346 5 Homogeneous Second-Phase Precipitation
Figure 5-26.Nucleation barrier DF* (a) and critical
radius R* (b) as a function of supersaturation at T=
0.25T
caccording to classical theory (cl. th.), non-
classical Cahn–Hilliard continuum model (C–H) and
discrete lattice model (DLM). R* is determined such
that it corresponds to (c
0+c
c)/2 where c
cis the com-
position of the nucleus at its center. (After Le Goues
et al., 1984a).www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

399. 5.5 Early Stage Decomposition Kinetics 347
of a coarsening process rather than a nucle-
ation event. By making certain assumptions
they attempted to infer the steady-state nu-
cleation rate from the particle density ob-
served during coarsening. The observed
lack of agreement between the experimen-
tally derived nucleation rates and the theo-
retically predicted ones may be seen to re-
sult from these experimental difficulties.
5.5.1.4 Non-Classical Nucleation –
Diffuse Interface Model
In the droplet model that classical nucle-
ation is based on, it is assumed that the
composition of the nucleus is more or less
constant throughout its volume and that its
interface is sharp. This made it possible to
take the change of volume Helmholtz en-
ergy and the interfacial Helmholtz energy
separately into account (Eq. (5-18)). In
non-classical nucleation theory, developed
by Cahn and Hilliard (1959a, b), the inho-
mogeneous solid solution in its metastable
state is considered to contain homophase
fluctuations with diffuse interfaces and a
composition which varies with position
throughout the cluster (Fig. 5-6 b). Hence,
unlike in the droplet model a critical fluctu-
ation has now to be characterized by at
least two parameters, its spatial extension
or wavelength 
land its spatial composition
variation. The necessary Helmholtz energy
formalism, which no longer treats volume
energy and surface energy separately, was
elaborated by Cahn and Hilliard (1958).
They wrote the Helmholtz energy change
associated with the transfer from the homo-
geneous system with composition c
0,
(5-33a)
to that of the inhomogeneous system with
(5-33b)
FfcKcV
V
=+∇∫[() *( )]
2
d
FfcV
V
00
=∫()d
as:
(5-33c)
The basic idea behind these expressions is to subdivide the solid with volume Vinto
many small volume elements dV, a proce-
dure which is often referred to as ‘coarse
graining’ (Langer, 1971). The free energy
of each volume element is taken asf(c)dV,
f(c) is considered to be the Helmholtz en-
ergy per unit volume of the bulk material with composition cwhich is equal to the
mean composition c(r) within dV located
at position r. In essence, coarse graining re-
quires the number of atoms within each volume element to be sufficiently large such that c(r), as well asf(c), can be spec-
ified as continuous functions. On the other hand, the number of atoms within dVmust
be small enough in order to avoid phase separation within the individual volume element. For the non-uniform system with composition fluctuations, Cahn and Hilli- ard (1958, 1959a, b) assumed that the local Helmholtz energy contains further terms depending on the composition gradient  (—c). They finally showed that the non-uni-
form environment of atoms in a composi- tion gradient may be accounted for by add- ing to the local Helmholtz energy a single gradient energyterm which is proportional
to (—c )
2
. Hence, the resulting Helmholtz
energy of a volume element dV is ex-
pressed asf(c)+K*(—c)
2
where the con-
stant K* denotes the gradient energy coeffi-
cient. Summing up all contributions from the various volume elements yields Eq. (5-33c).
The continuum model is base on the as-
sumption that the Helmholtz energyf(c)
varies smoothly with composition, i.e. the wavelength of the fluctuations must be large compared with the interatomic spac-
DFFF
fc fc K c V
V
=
=+∇
∫
(– )
[()– () *( )]
0
0
2
dwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

400. ing. Furthermore, for a continuum model
its is required that the two evolving phases
are fully coherent with each other and thus
have the same crystal structure with similar
lattice parameters. If the lattice parameter
changes with composition, which is com-
monly the case in crystalline solids, the re-
quirement for coherency leads to cohe-
rency strains which according to Eq. (5-12)
are accounted for by an elastic energy term
f
el(c)=h
2
Y(c–c
0)
2
(5-34)
Combining Eqs. (5-34) and (5-33c) yields
(5-35)
This expression may be interpreted in sim-
ilar terms as for classical nucleation [Eq.
(5-18)]: neglecting the elastic energy term,
the positive contribution of the gradient en-
ergy as a barrier to nucleation acts like the
surface Helmholtz energy in the droplet
model and is finally overcome by the gain
in chemical Helmholtz energy once the
composition difference between the fluctu-
ation and the homogeneous matrix has be-
come sufficiently large.
Assuming isotropy, the composition pro-
file c(r) of a spherical fluctuation (ris the
radial distance from the fluctuation center)
is obtained from a numerical integration of
(Cahn and Hilliard, 1959a,b):
(5-36)
with the boundary conditions dc/dr=0 at
the nucleus center (r≈0) and far away from
it (rÆ•) where c∫c
0. The critical nu-
cleus is then determined as the fluctuation
which, like a critical droplet, is in unstable
equilibrium (Eq. (5-35)) with the matrix.
Its composition is established such that 
the nucleation barrier (e.g., de Fontaine,
24
2
2
0
K
c
r
K
r
c
r
f
c
f
c
c
c
*
*
–
d
d
d
d
+=
∂
∂
∂
∂
DFfcfcKc
Yc c V
V
=+∇
+∫[()– () *( )
(– )]
0
2
2
0
2
h d
1982),
(5-37)
attains a minimum. Like in classical nucle-
ation, Df* is the vertical distance from the
tangent at c
0to the Helmholtz energy curve
at c* (cf. Fig. 5-5b). Suppose DF* is
known; the nucleation rate is then obtained from Eq. (5-27) or (5-29). Provided the constraint free energy is known, the com- position profile of a critical nucleus in a solid solution with composition c
0and gra-
dient energy coefficient K* (K* is of the or-
der 10 to 100 J m
–1
(at. fract.)
–2
) can be ob-
tained from integrating Eq. (5-36). Fig. 5-27 shows the composition profiles of nu- clei in supersaturated solid solutions with different solute concentrations c
0quenched
to T/T
c= 0.25, where T
cis the critical tem-
perature (LeGoues et al., 1984b); these cal- culations are based on the regular solution model forf(c) (cf. Sec. 5.2.4). For small
supersaturation (c
0=2¥10
–3
) the compo-
sition within the nucleus is constant and corresponds to c
b
eof the equilibrium b-
phase. The interface is almost sharp, simi- lar to that assumed in the classical droplet model. Correspondingly, the free energy of formation of the nucleus (DF*) and its size
(R*) as evaluated from either classical or
non-classical theory are almost identical (Fig. 5-26) and tend to infinity as c
0ap-
proaches c
a
e. With increasing supersatura-
tion the composition profile becomes in- creasingly diffuse and the solute concentra- tion at the nucleus center decreases (Fig. 5-27); this situation is no longer accounted for by the droplet model. Furthermore, as the initial composition (c
0) approaches the
spinodal one (c
s
a), unlike the droplet
model, non-classical theory predicts DF*
to go to zero (Fig. 5-26a), and the spatial
∆∆
∆
Fc r F
fc K
c
r
rr
[*()] *
(*) *
≡
=+
⎛
⎝
⎞
⎠
⎡
⎣
⎢
⎤
⎦
⎥
∞
∫4
2
0
2
p
d
d
d
348 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

401. 5.5 Early Stage Decomposition Kinetics 349
extent of the critical fluctuations (or R*) to
increase again; at c
0∫c
s
a, R* finally be-
comes infinite (Fig. 5-26 b) suggesting a
discontinuity of the decomposition mecha-
nism at the spinodal line (cf. Sec. 5.2.3).
This apparent discontinuity at the spino-
dal results from the assumption that the de-
composition path is controlled by the mini-
mum height of the nucleation barrier, i.e.,
by DF*, which close to the spinodal is in
the order of a few kT (Fig. 5-26a) and,
hence, rather low. As pointed out by de
Fontaine (1969), in a system close to the
spinodal many fluctuations exist, the spa-
tial extent of which is much smallerthan
that of the critical fluctuation; neverthe-
less, their formation energies are only
slightly higher than DF*. In this case, the
probability of the alloy decaying via the
formation of those ‘short-wavelength’
fluctuations requiring a slightly higher ac-
tivation energy is much higher than for a
decay via the formation of critical ‘long-
wavelength’ fluctuations.
5.5.1.5 Distinction Between Classical
and Non-Classical Nucleation
For a practical application of non-classi-
cal nucleation theory to experimental stud-
ies of unmixing, the total constraint free
energyf
¢(c) rather far away from equilib-
rium, as well as K*, have to be known in
order to calculate the composition profile
[Eq. (5-36)] and the nucleation energy [Eq.
(5-37)]. This is the case only for simple al-
loy systems and, hitherto, has inhibited the
application of the non-classical theory to
either scientifically or industrially impor-
tant alloy systems. Furthermore, so far it
has been assumed that a nucleus is in equi-
librium with the infinite matrix and that the
nuclei do notinteract during the nucleation
stage. This may be a valid approximation
as long as the supersaturation is small, i.e.,
where according to Figs. 5-26 and 5-27
classical nucleation theory applies. It ought
to become, however, a poor approximation
for more concentrated alloys where non-
classical nucleation applies. In this case the
nucleation rate is high (then the steady-
state nucleation regime will not be ob-
served) and, hence, the nucleus density
large. These complications in non-classical
nucleation make it desirable to provide the
experimentalist with a criterion for the ap-
plicability of classical nucleation theory
which is much easier to handle. Cahn and
Hilliard (1959a,b) suggest classical theory
will apply if the width lof the (diffuse)
interface is considerably smaller than the
Figure 5-27.Composition profile of a spherical nu-
cleus (center at r= 0) in four different metastable
solid solutions with composition c
0quenched to T=
0.25T
cinto the miscibility gap. The continuous line
was calculated from the Cahn–Hilliard continuum
model (Eq. (5-36)), the crosses represent calcula-
tions based on the discrete lattice point model (after
LeGoues et al., 1984a). The computations are based
on a regular solution model with the solubility limit
c
a
e= 0.37¥10
–3
and the spinodal composition c
a
s
=
6.41¥10
–2
, ais the lattice parameter of an assumed
f.c.c. lattice and lthe interfacial width.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

402. core of the nucleus where the composition
is about constant (Fig. 5-27).
If we assume that a perceptible nuclea-
tion rate is obtained for DF*≤25kT(in
the literature, DF*≈60kTis frequently
stated for the maximum barrier at which
nucleation becomes measurable; this value,
however, is much too high for homogene-
ous nucleation to occur in solids) and em-
ploys Eqs. (5-19) and (5-20), classical nu-
cleation theory may be applied if (Cahn
and Hilliard, 1959a, b):
(5-38)
For Cu–Ti (
s
ab= 0.067 J/m
2
; T= 623 K),
Fe–Cu (
s
ab= 0.25 J/m
2
; T= 773 K), and
Cu–Co (
s
ab= 0.17 J/m
2
; T= 893 K) this
means that the interfacial width has to be
smaller than ≈0.56, 0.33, and 0.42 nm, re-
spectively, and, hence, has to be rather
sharp. So far it has not been possible to
measure the composition profile across a
nucleus/matrix interface. However, analyt-
ical field ion microscopy of Cu–1.9 at.% Ti
aged at 350°C for 150 s revealed the Ti
concentration of particles with radii of only
≈1 nm to decrease from 20 at.% (corre-
sponding to Cu
4Ti, cf. Sec. 5.2.1) to that of
the matrix within one to two atomic (111)-
layers (von Alvensleben und Wagner,
1984). For the chosen aging conditions nu-
cleation was evaluated to terminate within
about 60 s (Kampmann and Wagner, 1984),
thus after aging for 150 s the analyzed par-
ticles must have already experienced some
growth beyond the original size of the nu-
cleus. Nevertheless, these results provide
rather good evidence that during nucleation
the Cu
4Ti clusters may be considered drop-
lets with sharp interfaces rather than spa-
tially extended (long-wavelength) fluctua-
tions with diffuse interfaces, even though
DF* is only about 10kT.
l
kT
2
4
75
1ps
ab
∫
Although some caution is advisable in
treating these tiny particles (critical nuclei in solid-state transformation are tens to hundreds of atoms in size) in terms of a continuum theory
1
, and assigning them a
macroscopic surface and thermodynamic bulk properties, classical nucleation theory seems to be appropriate for an estimation of the nucleation rate. This conclusion will be corroborated in Sec. 5.7.4.
5.5.2 Diffusion-Controlled Growth
of Nuclei from the Supersaturated Matrix
Suppose that after nucleation the stable
nucleusis embedded in a still supersatu-
rated matrix. As is illustrated in Fig. 5-28,
the particle will then be surrounded by a
concentration gradient which provides the
driving force for solute diffusion, and thus
gives rise to its growth.
The growth rate can be controlled either
by the rate at which atoms are supplied to
the particle/matrix interface by diffusion or
by the rate at which they cross the interface
(Tien et al., 1973). It may be rationalized
that for small particles the interface reac-
tion is likely to be the rate-controlling step
since the diffusion distances are rather
short; once the particles have grown to a
certain size, the matrix will be depleted
from solute atoms and the associated re-
duction of the driving force makes diffu-
sion likely to be the slower and, thus, rate-
controlling step (Shewmon, 1965). The
transition from one step to the other de-
pends upon the relative magnitudes of so-
lute diffusion and interface mobility.
After termination of nucleation it is com-
monly assumed that the mobility of the
interface is sufficiently high in order to al-
350 5 Homogeneous Second-Phase Precipitation
1
Many FIM studies of two-phase alloys (e.g., Cu–1
at.% Fe) revealed the shapes of even tiny clusters
with as little as about twenty atoms to be already
compact rather than ramified (Wagner, 1982).www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

403. 5.5 Early Stage Decomposition Kinetics 351
low the solute concentration c
Rat the
curved interface to achieve local equilib-
rium (Doherty, 1983). In this case diffusion
is the rate-controlling step for the growth
of stable coherent nuclei in a homogeneous
precipitation reaction. An analytical solu-
tion of the adequate field equation,
(5-39)
requires certain approximations (Zener,
1949). These were critically examined and
compared by Aaron et al. (1970) for the
diffusion-controlled growth of spherical
and platelet-shaped particles. Regardless of
the particular approximation used, both the
radius Rof a spherical precipitate and the
half-thickness Tof a platelet grow with time
according to a parabolic growth law as:
R=
l
i(Dt)
1/2
(5-40)
and
T=
l
j(Dt)
1/2
(5-41)
respectively. The rate constants 
l
iand l
jin-
crease with increasing supersaturation, or,
more specifically, with an increase in the
factor
(5-42)
k
ct c
cc
*
()–
–
=2
R
pR
Dcrt
crt
t
∇=
∂
∂
2
(,)
(,)
in a manner which, in particular for larger values of k*, depends upon the approxima-
tion assumed (Aaron et al., 1970). For pre- cipitating intermetallic compounds, k* is
one or two orders of magnitude larger (e.g., ≈0.3for 
g¢-Ni
3Al in Ni–Al and ≈0.2for
b¢-Cu
4Ti in Cu–Ti) than for precipitating
phases with c
p≈1such as Fe–Cu, Cu–Fe
and Cu–Co.
For small supersaturations, c(r,t) in Eq.
(5-39) may be approximated as being time- independent. In this case, an isolated spher- ical particle with radius Rsurrounded by
the concentration field illustrated in Fig. 5-28 will grow at a rate
(5-43)
where D(assumed to be independent of
composition) is the volume diffusion coef- ficient in the matrix. According to the Gibbs–Thomson equation, the composition c
Rof the matrix phase at a curved interface
is different from that of a flat interface, the latter being in equilibrium with the equilib- rium solute concentration c
a
eof the a-ma-
trix phase, and varies with the precipitate radius as (Martin and Doherty, 1976)
(5-44)
with 
g
a(c
R) and g
a(c
a
e) being the activity
coefficients of the solute atoms in the a-
phase at the concentration c
Rand c
a
e, re-
spectively. If the solid solution shows reg- ular solution behavior (
g
athen becomes in-
dependent of c), Eq. (5-44) yields the well-
known Gibbs–Thomson equation:
(5-45)
or for larger radii in its linearized version,
(5-46)
cR c
V
RT R
R
e
g()=+
⎛
⎝
⎜
⎞
⎠
⎟
a
ab b1
2 1
s
cR c
V
RT R
R
e
g( ) exp=
⎛
⎝
⎜
⎞
⎠
⎟
a
ab b
2 1s
ln
() –
–
–ln
()
()
cR
c
c
cc RT
V
R
c
c
R
e
e
p
e
g
R
e
a
a
a
ab b a
aa
=
⎛
⎝ ⎜
⎞
⎠ ⎟
1 2
s g
g
d
d
R
pRR
t
R
ct c
cc
D
R
≡=v()
()–
–
Figure 5-28.Schematic concentration field in the
matrix surrounding a nucleus with radius Rand com-
position c
p.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

404. By assuming a monodispersive particle
distribution and c
R≈c¯(tÆ•)=c
a
e(or c¢
a
e),
and c¯(t)≈c
0, integration of Eq. (5-43) yields
(5-47)
It should, however, be pointed out that, so
far, there is little experimental evidence for
the existence of a precipitation regime dur-
ing which particle growth strictly follows
Eq. (5-47) (Kampmann and Wagner, 1984).
In Sec. 5.7.4.3 the reason for this lack of
experimental evidence will be provided, as
well as a guideline to the design of experi-
ments that allow a verification of the exis-
tence of diffusion-controlled particle
growth with the predicted kinetics to be
made. Most quantitative experimental ob-
servations of growth rates which yielded
good agreement with the diffusion-con-
trolled models outlined above, were con-
fined to large particles, frequently with
sizes in the micrometer range (Aaronson et
al., 1970a, b; Doherty, 1982).
5.5.3 The Cluster-Dynamics Approach
to Generalized Nucleation Theory
As has been pointed out in Sec. 5.5.1.1,
the cluster-dynamics approach chosen by
Becker and Döring (1935) is based on the
assumption that the equilibrium cluster dis-
tribution function can be specified by Eqs.
(5-23) and (5-18). Even though the result-
ing predictions of classical nucleation the-
ory ought to become valid asymptotically
for large droplets at small supersaturations,
there remain some inherent deficits 
(Binder, 1980; Gunton et al., 1983).
For instance, for small clusters the sur-
face energy entering Eq. (5-18) should
contain size-dependent corrections to the
macroscopic equilibrium energy of a flat
interface. Furthermore, the separation of
the droplet formation Helmholtz energy
Rt
cc
cc
Dt()
–
–
()
/
/
=
⎛
⎝
⎜
⎞
⎠
⎟
2
0
12
12a
a
e
p
e
into a bulk and a surface term appears rather debatable. To avoid some of these deficits, there have been several attempts to develop more accurate descriptions of clusters, tak- ing into account their different sizes and shapes (e.g., Binder and Stauffer, 1976; Stauffer, 1979), and to derive an equilibrium distribution of clusters which is more realis- tic than the one given by Eq. (5-23). As in the classical nucleation theory, the latter is an important quantity for a computation of the cluster formation rate in terms of the more recently elaborated theories of cluster dynamics which, in essence, are extensions of the Becker–Döring theory (Binder and Stauffer, 1976; Penrose and Lebowitz, 1979; cf. Gunton et al., 1983 and the chapter by Binder and Fratzl (2001) for a general discussion of the various developments).
Unlike Becker and Döring, however,
who confined the microscopic mechanism of cluster growth or shrinkage to the con- densation or evaporation of monomers (Sec. 5.5.1.1), in their generalized nuclea-
tion theoryBinder and coworkers consider
the time evolution of the cluster size distri- bution functionf(i,t) more generally in
terms of a cluster coagulation or cluster
splitting mechanism, i.e., in a single cluster
reaction step i-mers (i= 2, 3, …) are also
allowed to be added or separated from an existing cluster (e.g., Binder, 1977; Mirold and Binder, 1977). Thus, the related kinetic equation for the time evolution off(i,t) 
(Binder, 1977),
(5-48)
contains four different terms. The first ac-
counts for an increase of i-mers due to
d
dt
fit fi i t
fit
fi t fi i t fit fi t
iii
i
ii
i
i
iii
i
i
ii
i(, ) ( , )
–(,)
(,)(–,)– (,)(,
)
,
,
–
–,
–
,=+ ′
+
×′′ ′
+′′
′=
∞
′
′=
′′
′=
′
′=
∞∑
∑∑
∑a
ab
b
1
1
1
1
1
1
1
2
1
2
352 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
+ s e  l ∫'4 , kp e  r  i ∫&s ! 9 j+  N   0 e

405. 5.5 Early Stage Decomposition Kinetics 353
splitting reactions (i+i¢)Æ(i,i¢) which
are assumed to be proportional to the mo-
mentary number f(i+i¢,t) of clusters of
size (i+i¢), where the proportionality co-
efficient a
i+i¢,i¢ is the rate constant. The
second term describes the decrease of 
i-mers because of splitting reactions iÆ
(i–i¢,i¢); as a reverse process, coagulation
of clusters with sizes (i–i¢) and i¢contrib-
utes to a further increase of i-mers (third
term), some of which are lost again by coag-
ulation reactions between i-mers and i¢-
mers to yield (i+i¢)-mers (fourth term). 
Evidently, for i¢= 1 the monomer evapora-
tion and condensation process, assumed to
be the rate controlling step in the Becker–
Döring theory, is contained in Eq. (5-48).
If, again as in classical nucleation (Sec.
5.5.1.1, Eq. (5-24)), detailed balance con-
ditions are assumed to apply between split-
ting and coagulation, then the rates 
aand b
in Eq. (5-48) can be replaced by a single
reaction rate W, e.g.,
(5-49)
W(i,i¢)∫a
i+i¢,i¢ C(i+i¢)= b
i,i¢C(i)C(i¢)
Here, C(i) again denotes the cluster con-
centration which is in thermal equilibrium
with the metastable matrix.
Numerical integration of Eq. (5-48) with
Eq. (5-49) provides the time evolution of
the cluster concentration. This requires,
however, a knowledge of the initial cluster
distributionf(i,t= 0), the reaction rates
such as W(i,i¢), and of C(i). These quan-
tities are not commonly available for alloys
undergoing phase separation; this has so
far impeded a quantitative comparison
between the predictions of the generalized
nucleation theory and experimental results
on nucleation kinetics.
However, the solution of the kinetic
equations with plausible assumptions for
the missing quantities (Mirold and Binder,
1977, see also the chapter by Binder and
Fratzl (2001) has provided a profound,
though basically qualitative, insight into
the dynamics of cluster formation and
growth. As is shown in Fig. 5-29, the in-
itially (t= 0) monotonically decreasing
cluster size distribution attains for interme-
diate times a minimum at a size that corre-
sponds to about that of the critical nucleus
in classical nucleation theory.
The broad maximum off(i,t) (which,
in fact, does not appear in the correspond-
ing curve of the Becker–Döring nucleation
theory (cf. Fig. 5-24), and its shift to larger
sizes with increasing time is due to those
particles which have nucleated at earlier
times and, hence, have already experienced
growth. At tÆ•equilibrium is reached.
In this case large clusters (iÆ•) of the 
precipitated equilibrium phase with com-
position c
b
eare embedded in a matrix 
(with composition c
a
e) which then still con-
tains smallclusters with the equilibrium
size distribution shown in Fig. 5-29.
Figure 5-29.Cluster size distribution at different
aging times (arbitrary units) as obtained from a nu-
merical solution of Eqs. (5-48) and (5-49). (After
Binder and Stauffer, 1976.)www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

406. The qualitative features of thef(i,t)-
curves emerging from the generalized nu-
cleation theory and displayed in Fig. 5-29
are corroborated by computer simulations
(cf. Sec. 5.5.6). They are indicative of the
occurrence of nucleation and growth as
concomitant processes during the early
stages of unmixing. For most solids under-
going phase separation, therefore, it is not
possible to investigate experimentally nu-
cleation and growth as individual processes
proceeding subsequently on the time scale.
Apart from well designed experiments, it is
thus usually impossible to verify experi-
mentally the kinetics predicted by classical
or non-classical nucleation theory (Secs.
5.5.1.1 to 5.5.1.3) or predicted by growth
theories (Sec. 5.5.2). This will be substan-
tiated in more detail in Sec. 5.7.4.
The generalizing characterof the nucle-
ation theory of Binder and coworkers is
founded on the fact that, in the sense of
Sec. 5.2.3, it comprises nucleation in the
metastable regime as well as the transition
to spinodal decomposition in the unstable
regime of the two-phase region. Conse-
quently, the artificial divergency of both
the critical radius R* of the nucleus and the
wavelength 
l* of a critical fluctuation, in-
herent to the Cahn–Hilliard theories of non-
classical nucleation (Sec. 5.5.1.4) and spi-
nodal decomposition (Sec. 5.5.4), is no
longer discernible on approaching the spi-
nodal line either from the metastable 
(‘nucleation’) or from the unstable 
(‘spinodal’) regime of the two-phase re-
gion (Fig. 5-30). On crossing the spinodal
curves the size of a critical cluster de-
creases steadily until it becomes compar-
able to the correlation length of typical
thermal fluctuations (Binder et al., 1978).
Thus, no discontinuity of the mechanism or
of the decomposition kinetics is expected
to occur on crossing the border between the
metastable and unstable regions.
From the experimental point of view it is
rather difficult and tedious to obtain statis-
tically significant information from direct
imaging techniques (FIM, HREM) on the
time evolution of the cluster size distribu-
tion during the earliest stages of unmixing.
Attempts were made by Si-Qun Xiao and
Haasen (1991) who employed HREM to
binary Ni–Al alloys. This information,
however, is implicitly contained in the
structure function S(
k,t) from small angle
354 5 Homogeneous Second-Phase Precipitation
Figure 5-30.a) Schematic phase diagram of a bi-
nary alloy (components A and B) with a symmetrical
miscibility gap.
b) Variation of the critical radius R* of a nucleus and
of the wavelength 
l
cof a critical fluctuation with
composition of the alloy at T
0according to the non-
classical Cahn–Hilliard nucleation theory (C–H non-
cl.) and the Cahn–Hilliard spinodal theory (C–H
spin.). Approaching the spinodal composition c
a
s
from either the metastable or the instable region
causes R* and, respectively, to diverge. In contrast,
the generalized nucleation theory of Binder and co-
workers yields the size of the critical cluster to de-
crease steadily until it becomes comparable with the
correlation length of typical thermal fluctuations.
There is no peculiarity on crossing the spinodal
curve.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

407. 5.5 Early Stage Decomposition Kinetics 355
scattering techniques, e.g., SAXS or SANS
(Sec. 5.3.1.2). Binder et al. (1978) com-
puted the time evolution of S(
k,t) on
grounds of their cluster dynamics model.
For this purpose, rather crude approxima-
tions and simple assumptions had to be
made concerning the originally complex
shape of the size distribution function of
the clusters, their shapes and their compo-
sition profiles. Hence, the resulting S(
k,t)
is only qualitative in nature and cannot be
used for a quantitative comparison with ex-
perimentally obtained structure functions.
As is shown in Fig. 5-31a, S(
k,t) is pre-
dicted to develop a peak the height S
m∫
S(
k
m) of which increases with time due to
the increasing cluster volume fraction,
while the peak position is shifted towards
smaller values of 
k, reflecting the coarsen-
ing of the cluster size distribution with
time. Fig. 5-31b shows the variation of
both the peak height S
m(k∫k
m) and the
peak position 
k
mwith time. On a double-
logarithmic plot both curves display som
curvature, though the shift of 
k
mmay be
reasonably well represented as a power law,
k
m(t)µt
–1/6
(5-50)
during the ‘aging period’ covered in Fig.
5-31a and 5-31b.
Even though it is not feasible to con-
vert the scaled times shown in Fig. 5-31a
and 5-31b to real times, experimental
SANS curves (Fig. 5-32a) and the related
S
m(t)µdS
m
(t)/dW(cf. Eq. 5-17)) and
k
m(t)-curves (Fig. 5-32 b) from Cu–2.9
at.% Ti homogenized at T
H, quenched and
isothermally aged at 350°C (Eckerlebe et
al., 1986) display the characteristic fea-
tures qualitatively predicted by the gener-
alized nucleation theory.
Kampmann et al. (1992) developed a ge-
neric cluster-dynamics model that can be
used for the quantitative interpretation of
experimental kinetic data.
Unlike previous models, in this model a
distinction can be made between the clus-
tering phase being a disordered solid solu-
tion, e.g., Cu–Co with a positive pair ex-
change energy, or an ordered phase, e.g.,
Figure 5-31.a) Time evolution of the structure func-
tion S(
k,t), and b) of its peak height S
mand peak po-
sition 
k
mas predicted by the generalized nucleation
theory of Binder and coworkers for a 3-dimensional
Ising model (c
0=0.1, T/T
c= 0.6). After Binder et al.
(1978), see also the chapter by Binder and Fratzl
(2001).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

408. Ni
3Al in the Ni–Al system with a negative
pair exchange energy (Staron and Kamp-
mann, 2000a, b). In the latter case the con-
tinuation of the atomic order must be ac-
counted for when computing the growth
probability of the long-range ordered
Ni
3Al cluster, the growth of which is as-
sumed to proceed via diffusional incorpo-
ration of a solute monomer (cf. Sec. 5.7.7).
356 5 Homogeneous Second-Phase Precipitation
Figure 5-32.a) Time evolu-
tion of the structure function 
(SANS experiments) of a
Cu–2.9 at.% Ti single crystal
aged at 350 °C for the given
times. b) Time evolution of the
peak height S
mµdS
m
/dWand
the peak position 
k
m. (After
Eckerlebe et al., 1986.) 
5.5.4 Spinodal Theories
As has been pointed out in Sec. 5.2.3 the
evolution of non-localized, spatially ex-
tended solute-enriched fluctuations into
stable second phase particles is treated in
terms of spinodal theories. The general
concepts of the spinodal theories, including
more recent extensions, are comprehen-
sively discussed by Binder and Fratzl
(2001) in Chapter 6 of this volume. We will
therefore restrict ourselves to a summary of
the essential predictions of those theories
which may be examined by experimental
studies. Although often not justified, in ma-
terials science some of these predictions are
frequently employed as sufficient criteria to
distinguish a spinodal reaction from an un-
mixing reaction via nucleation and growth.
In their continuum model for spinodal
decomposition, which is based on the free
energy formalism of a non-uniform binary
alloy outlined in Sec. 5.5.1.4 (Eq. (5-35)),
Cahn and Hilliard (1959a, b) derived the
following (linearized) diffusion equation:
(5-51)
∂
∂
=∇
∂
∂
⎛
⎝
⎜
+
⎞
⎠
⎟
⎡
⎣
⎢
⎢
×∇
⎤
⎦
⎥
ct
t
M
n
f
c
Y
ct K ct
c
(,)
(,)– * (,)
r
rr
v
2
2
2
2
4
0
2
2hwww.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

409. 5.5 Early Stage Decomposition Kinetics 357
for the time dependence of the composition
c(r,t) at position r. As the number of at-
oms per unit volume, n
v, accounts for the
fact that the derivative has to be taken with
respect to the concentration of component
B, Mis the atomic mobility and is related to
the interdiffusion coefficient D
˜
via
(5-52)
The other symbols in Eq. (5-51) were de-
fined in Sec. 5.5.1.4.
As Mis always positive, D
˜
takes the sign
of ∂
2
f/∂c
2
and so is negative inside the spi-
nodal regime (Sec. 5.2.2), giving rise to an
‘uphill diffusion’ flux of solute atoms (cf.
Fig. 5-6 b).
The linearized version of the more gen-
eral nonlinear diffusion equation holds if
the amplitude c(r)–c
0of the composition
fluctuation is rather small and both Mand
(∂
2
f/∂c
2
)|
c
0
are independent of composi-
tion. These approximations inherently con-
fine the Cahn–Hilliard theory (CH theory)
to the earliest stages of phase separation. 
By expanding the atomic distribution
c(r,t) into a Fourier series, Eq. (5-51) can
be written as
(5-53)
with the amplitude
(5-54)
of a Fourier component with wavenumber
k=2p/lif ldenotes the wavelength of the
composition fluctuation. Eq. (5-51) has a
solution for every A(
k,t) with
(5-55)
where A(
k, 0) is the initial amplitude of
the Fourier component with wavenumber
k. The so-called amplification factor R( k)
AtA Rt( , ) ( , ) exp [ ( ) ]kkk kk k= 0
Acci( ) [ ( ) – ] exp ( )kkk k=∫rrr
0 d
∂
∂
=
A
t
RA
()
() ()kk
kkkk
MDn
f
c
c
≡
∂
∂
˜
v
2
2
0
is defined as:
(5-56)
With reference to Eqs. (5-16), (5-54) and
(5-55), the structure function S(
k,t) or the
related small angle scattering intensity (Eq. (5-17)) at time tis obtained as
(5-57)
where S(
k, 0) denotes the equal-time
structure function of the system equili- brated at the homogenization temperature.
Inside the spinodal region (∂
2
f/∂c
2
)|
c
0
is
negative. Thus for a given negative value of the latter, R (
k) becomes positive for all
wavenumbers 
ksatisfying
(5-58 a)
or, correspondingly, for all fluctuation wavelengths 
llarger than l
cwith
(5-58 b)
Long wavelength fluctuations with 
l>l
c
thus will be amplified exponentially whereas short wavelength fluctuations will decay with aging time. For 
l
m∫∂∫2 l
c, R(l)
attains a maximum giving rise to fastest growth of fluctuations with wavelength 
l
m.
On approaching the coherent spinodal curve where the denominator in Eq. (5-58b) becomes zero (cf. Sec. 5.5.3), the critical wavelength 
l
cdiverges in a similar
manner as does the critical radius R* (Fig.
5-30). Thus both the linear theory on spino- dal decomposition and the non-classical nucleation theory (Sec. 5.5.1.4) predict abrupt changes in the decomposition kinet- ics on crossing the spinodal line from ei- ther side. This has prompted several ex- perimental studies to determine the spino-
lp h
c
2
2
2
242
0
=
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
K
f
c
Y
c
*–
kk h
2
2
2
2
0
22<=
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟c
2
f
c
YK
c
–*
StS Rt( , ) ( , ) exp [ ( ) ]kkk kk k= 02
R
M
n
f
c
YK
c
() – *kk=
∂
∂
⎡
⎣
⎢ ++
⎤
⎦
⎥
⎥
v
2
2
222
0
22hkkwww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

410. dal curve on grounds of a kinetic distinction
between metastable and unstable states.
In terms of the time evolution of the struc-
ture function [Eq. (5-57); now assumed to be
isotropic] or the small angle scattering inten-
sity, S(
k,t) should increase exponentially for
k<k
cwith a peak at the time-independent
position 
k
m=k
c/∫≤2. Furthermore, all S( k)
curves taken at various times should cross at
a common cross-over point,at 
k
c.
Most SAXS or SANS studies of binary
alloys, oxides or glasses that were deeply
quenched into the miscibility gap failed to
corroborate the predictions of the linear-
ized CH theory. This is exemplified by
Cu–2.9 at.% Ti isothermally aged at
350°C. As illustrated in Fig. 5-32b, the
peak position of the structure function,
which is frequently identified with the
mean cluster spacing l
–
(l
–
≈2
p/k
m=l
m), is
not found to be time-independent but is
rather shifted towards smaller values of 
k
owing to spontaneous coarsening of the
clustering system. Furthermore, nocom-
mon cross-over point at any 
k
c(Fig. 5-32a)
exists nor does S(
k,t) grow exponentially
for any value of 
k(Fig. 5-33).
According to Eqs. (5-56) and (5-57), a
plot of
(5-59)
versus 
k
2
should yield a straight line with 
R(
k) ∫0 at k
c, whereas a pronounced cur-
vature at larger values of 
khas been ob-
served experimentally in alloys (e.g., Fig.
5-34) and glass systems (Neilson, 1969).
Cook (1970) attributed this curvature to
random thermal composition fluctuations
which were not accounted for in the origi-
nal CH theory. Even with the incorporation
of the thermal fluctuations into the linear-
ized theory, at its best the resulting
Cahn–Hilliard–Cook (CHC) spinodal the-
ory is seen (Langer, 1973; Gunton, 1984)
to be valid only for the earliest stages of
decomposition in systems for which the
range of the interaction force is consider-
ably larger than the nearest-neighbor dis-
tance, i.e., in systems which are almost
mean-field-like in nature. This is not the
case for metallic alloys, oxides and glasses,
but is for polymer blends. In fact, results
R
t
StS( )/ ( / ) ln [ ( , )/ ( , )]kk k k k
22
12 0=
∂
∂
358 5 Homogeneous Second-Phase Precipitation
Figure 5-33.Time evolution of d S/dWfor Cu–2.9
at.% Ti aged at 350°C for various constant wave-
numbers 
k(Eckerlebe et al., 1986).
Figure 5-34.Variation of R( k,t)/k
2
with k
2
as de-
termined for two different time intervals (Eckerlebe et al., 1986). www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

411. 5.5 Early Stage Decomposition Kinetics 359
from kinetic studies of decomposition in
various polymer mixtures, some of which
are cited in Table 5-1 (for reviews, cf. Gun-
ton et al., 1983; Binder and Fratzl (2001),
Chapter 6 of this volume), were found to be
consistent with the predictions of the line-
arized spinodal CHC theory (e.g., Okada
and Han,  1986). In contrast, apart from a
SANS study on the kinetics of phase separ-
ation in Mn–33 at.% Cu (Gaulin et al.,
1987), none of the scattering experiments
on metallic alloys or oxides which are listed
in Table 5-1and which probably had been
quenched into the unstable region of the
miscibility gap, corroborated the predic-
tions of the CHC theory. SANS curves
taken from Mn–33 at.% Cu during short-
term aging (between 65 s and 210 s at
450°C) could be fitted to the CHC structure
function with 
k
mbeing time-independent
during this period. Thus, it was concluded
that phase separation in Mn-33 at.% Cu is of
the spinodal type and the kinetics follow the
CHC predictions (Gaulin et al., 1987). The
S(
k) curve of the as-quenched state already
revealed a peak, in fact, at a smaller value of
k
mthan for the aged samples. This is indic-
ative of the fact that phase separation oc-
curred during the quench owing to a finite
quench rate. In this case, as has been exten-
sively discussed by Hoyt et al. (1989), 
k
m
may indeed become initially time-indepen-
dent, thus pretending an experimental con-
firmation of the CHC linear approximation.
In the framework of the mean-field theo-
ries the spinodal curve is well defined as
the locus where ∂
2
f/∂c
2
vanishes (Sec.
5.2.2). It can then be easily evaluated, e.g.,
from an extrapolation of the thermody-
namic data which are known for the single-
phase solid solution, into the (‘mean-field’)
spinodal regime (Hilliard, 1970). Since,
however, neither metallic alloys nor glass
or oxide mixtures behave as mean-field-
like systems, no uniquely defined spinodal
curve exists for these solid mixtures (cf.
Binder and Fratzl (2001), Sec. 6.2.5, this
volume). Thus in general, it is currently not
feasible to predict on theoretical grounds
whether a metallic, glass or oxide solid
mixture has been quenched into the meta-
stable or unstable region of the miscibility
gap. In this context, it remains debatable
whether each of the metallic, glass or oxide
mixtures that were studied by means of
SAXS or SANS, and which have been re-
ported to undergo spinodal decomposition
(Table 5-1), were truly quenched into and
aged within the spinodal region of the
phase diagram. Nevertheless, extensive ex-
perimental SAXS or SANS studies, in par-
ticular on Al–Zn alloys and Fe–Cr (Table
5-1), did not reveal any evidence for a dras-
tic change in the time evolution of the S (
k,t)
curves with a change in the initial supersat-
uration. For the Al–Zn systems in particu-
lar, it was shown (Simon et al., 1984) that
the clustering rate increases rapidly with
increasing supersaturation, i.e. both 
k
mand
S
mare larger the deeper the quench. 
Extensions of the Cahn–Hilliard theory
of spinodal decomposition to ternary or
multicomponent systems have been elab-
orated by de Fontaine (1972, 1973) and
Moral and Calm (1971). However, to our
knowledge the kinetic predictions from
these extensions have not yet been com-
pared with experimental results, probably
because of difficulties in determining the
partial structure functions of ternary alloys
(cf. Sec. 5.3.1.2).
As has been pointed out in Sec. 5.2.3,
the formulation of a ‘unified theory’ com-
prising both nucleation and growth as well
as spinodal decomposition can also be at-
tacked on grounds of a spinodal theory.
This has been attempted in the statistical
model of Langer, Bar-On and Miller
(1975), which takes into account thermal
fluctuations and nonlinear terms which arewww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

412. 360 5 Homogeneous Second-Phase Precipitation
Table 5-1.Some small angle scattering experiments – sometimes jointly employed with other microanalytical
tools for studying the morphology – primarily designed for an investigation of spinodal decomposition. If not
otherwise stated the concentrations are in at.% (N.d.: not determined.)
Decomposing solid Experimental Morphology of two- Authors
technique phase microstructure
Metallic alloys
Al–4 wt.% Cu SAXS Modulated structure Naudon et al. (1976)
Al–22% Zn SAXS Gerold and Merz (1967)
Al–22% Zn SAXS, TEM Modulated structure Agarwal and Herman (1973)
Al–20.7 … 49.1% Zn SAXS Bonfiglioli and Guinier (1966)
Al–5.3 … 6.8% Zn SANS GP-zones Hennion et al. (1982)
Al–5.3 … 12.1% Zn SANS, TEM Guyot and Simon (1981)
Simon et al. (1984)
Al–22% Zn–0.1 Mg SAXS, TEM Interconnected spherical Forouhi and de Fontaine
clusters at early times; (1987)
regularly spaced platelets
at later stages
Al–12 … 32 Zn SAXS (synchro- Hoyt et al. (1987, 1989)
tron radiation)
Al–2.4% Zn–1.3 Mg SANS Blaschko et al. (1982, 1983)
Al–3.8 … 9% Li with SANS N.d. Pike et al. (1989)
additions of Cu, Mn
Au-60% Pt SANS Singhal et al. (1978)
Fe-28% Cr–10% Co SANS, TEM, AFIM ‘Sponge-like’ structure Miller et al. (1984)
Fe–34% Cr SANS N.d. Katano and Iizumi (1984)
Fe–30 … 50% Cr SANS N.d. Ujihara and Osamura (2000)
Fe–29.5% Cr–12.5% Co Anomalous SAXS N.d. Simon and Lyon (1989)
(synchrotron source)
Fe–52% Cr SANS N.d. La Salle and Schwartz (1984)
Fe–20 … 60% Cr SANS N.d. Furusaka et al. (1986)
Cu–2.9% Ti SANS, TEM Modulated structure Eckerlebe et al. (1986)
CuNiFe SANS Modulated structure Aalders et al. (1984)
CuNiFe SANS, TEM, AFIM Mottled structure Wagner et al. (1984)
CuNiFe Anomalous SAXS N.d. Lyon and Simon (1987)
(synchrotron source)
Cu–2% Co SANS N.d. Steiner et al. (1983)
Mn–25 … 52% Cu SANS (analysis of N.d. Vintaikin et al. (1979)
integrated intensity)
Mn–33% Cu SANS N.d. Gaulin et al. (1987)
Ni–13% Al SANS Anisotropic clustering Beddoe et al. (1984)
Ni–12.5% Si SANS, SAXS, TEM Modulated structure Polat  et al. (1986, 1989)
(‘side bands’)
Ni–11.5% Ti SANS, TEM Modulated structures Cerri et al. (1987, 1990)www.iran-mavad.com
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413. 5.5 Early Stage Decomposition Kinetics 361
neglected in the CH theory. The LBM non-
linear theory of spinodal decomposition
also accounts for the later stages of phase
separation, though notfor the coarsening
regime. If coherency strains are neglected
it approximates the kinetic evolution of the
structure function as
(5-60)
d
d
v
v
St
t
M
n
K
f
c
SAt
M
n
kT
c
(,)
–
*()*()k
k
kk
k
=
×+
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
+
2
2
2
2
2
2
2
0
With certain approximations, nonlinear ef-
fects and thermal fluctuations are con-
tained in the term A*(t) and in the last
term, respectively. Setting A*(t) ∫0 repro-
duces the corresponding equation of mo-
tion of the CHC theory, and, if the fluctua-
tion term is also omitted, Eq. (5-57) of the
original linear CH theory is regained.
Langer et al. (1975) proposed a computa-
tional technique for solving Eq. (5-60).
This required several approximations to be
made, e.g., on f(c) and on M , which inher-
ently leaves the LBM approach with some
fundamental shortcomings. Nevertheless,
with respect to the features displayed in the
Table 5-1.Cont.
Decomposing solid Experimental Morphology of two- Authors
technique phase microstructure
Glasses, oxides
a
B
2O
3–(15 wt.% PbO SAXS Craievich (1975)
–5 wt.% Al
2O
3) Acuña and Craievich (1979)
(quasi-binary system) Craievich and Olivieri (1981)
B
2O
3–(27 wt.% PbO SAXS Craievich et al. (1986)
–9 wt.% Al
2O
3) (synchrotron source)
Vycor glass SANS ‘Sponge-like’ structure Wiltzius et al. (1987)
consisting of a SiO
2-rich
and a B
2O
3-alkali
oxide-rich phase
SiO
2–13 mole% Na
2O SAXS, TEM ‘Sponge-like’ structure Neilson (1969)
TiO
2–(20–80 mole%) SnO
2SAXS, TEM, X-ray Lamellar modulations Park et al. (1976)
diffraction (‘side along [001] (Fig. 5-38)
band’ analysis)
Polymer mixtures
Critical mixture of per- Light scattering Wiltzius et al. (1988)
deuterated and protonated  Bates and Wiltzius (1989)
1,4-polybutadiene
polybutadiene and Izumitani and Hashimoto (1985)
styrene–butadiene
copolymer mixtures
polystyrene–polyvinyl- Light scattering Sato and Han (1988)
methylether (PS–PVME) Sato et al. (1989)
Snyder et al. (1983a, b)
Hashimoto et al. (1986a, b)
Okada and Han (1986)
a
For a comprehensive survey on oxides and glasses up to 1978, see Jantzen and Herman (1978).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

414. 362 5 Homogeneous Second-Phase Precipitation
Figure 5-35.a) Phase diagram of a binary alloy with
a symmetric miscibility gap centered at c
sym.
b) Time evolution of the structure function of an al-
loy with c
0=c
symquenched and aged at T
A. The in-
sert shows the distribution function of composition
configurations at two different times. At t= 80 the
evolving two-phase structure with compositions c
1
and c
2or y
1and y
2, respectively, with
already becomes discernible. All units are dimen-
sionless.
c) LBM predictions on the variation of R(
k,t)/k
2
with k
2
as determined from b) according to Eq.
(5-59). Qualitatively there is good agreement with
the LBM predictions and experimental results, e.g.,
displayed in Fig. 5-34. (After Langer et al., 1975.)
y
cc
cc
y
cc
cc
1
1
2
2==
–
–
–
–
sym
e
sym
sym
e
sym
and
a b
Figure 5-36.Time evolution
of the structure function  (SANS intensity) of Fe–40 at.% Cr aged at 515°C for the given times. Full lines are calculated from the LBM the- ory with three free fitting pa- rameters. (After Furusaka et al. 1986.)www.iran-mavad.com
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415. 5.5 Early Stage Decomposition Kinetics 363
time evolution of S( k,t), the predictions of
the LBM theory (Fig. 5-35) are in rather
good agreement with both experimental
studies (e.g., Fig. 5-32 a and 5-34) and
Monte Carlo studies (Sec. 5.5.6, Fig. 5-
41). 
In their study of phase separation in
Fe–(20, 30, 40 and 60) at.% Cr alloys, Fu-
rusaka et al. (1986) compared the experi-
mental S(
k,t) curves with Eq. (5-60) of
the LBM theory. For shorter aging times at
515°C, the experimental data points taken
from Fe–40 at.% Cr could well be fitted by
Eq. (5-60) (Fig. 5-36) if kT/K*, MK*, and
A*/K* were employed as three independent
fitting parameters. On the grounds of the
agreement with the spinodal theory of
LBM, and additionally, as this particular
alloy was evaluated to lie within the 
(mean-field) spinodal region, it has been
concluded that Fe–40 at.% Cr is a ‘spino-
dal alloy’.
Based on a similar LBM analysis of
SANS data from Fe–30 … 50 at.% Cr, Uji-
hara and Osamura (2000) corroborated this
conclusion recently.
The nonlinear theory of spinodal decom-
position developed by Langer (1971) for
binary alloys has also been extended to ter-
nary substitutional systems (Hoyt, 1989)
by deriving the time-dependent behavior of
the three linearly independent partial struc-
ture functions (cf. Sec. 5.3.1.2). An experi-
mental examination of the kinetic predic-
tions, however, is still lacking.
5.5.5 The Philosophy of Defining
a ‘Spinodal Alloy’ –
Morphologies of ‘Spinodal Alloys’
Referring to the previous section it is
rather difficult or even impossible to assess
on a thermodynamic basis whether an al-
loy, glass or oxide system was truly
quenched into and aged within the spinodal
region of the miscibility gap. Furthermore,
the various spinodal theories are difficult to
handle and many parameters in the result-
ing kinetic equations are often not avail-
able for most solid mixtures. Hence, like
the non-classical nucleation theory, the
various elegant theories that describe the
kinetics of phase separation of a ‘spinodal
alloy’ are of little use for the practical met-
allurgist.
Because of these problems, materials
scientists have up to now employed mor-
phological criteria for the definition of a
‘spinodal alloy’. These are simply related
to the predictions of the linear spinodal CH
theory concerning the morphological evo-
lution of a solid mixture undergoing spino-
dal decomposition.
Most crystalline solid solutions show a
variation of lattice parameter with compo-
sition leading to coherency strains. The as-
sociated strain energyf
el=h
2
Y(c–c
0)
2
(Eq. (5-34)), which is accounted for in Eq.
(5-51), reduces the driving force for phase
separation. This effect shifts the locus of
the original chemical(mean field) spinodal
to lower temperatures, yielding the coher-
ent spinodal curve (Eq. (5-58)). If the pa-
rameter Y(cf. Sec. 5.2.4), which is a com-
bination of various elastic constants, de-
pends on the crystallographic direction,f
el
also becomes anisotropic. Therefore, the
locus of the coherent spinodal may also
vary with the crystallographic direction.
This becomes particularly discernible for
the tetragonal TiO
2–SnO
2oxide system 
(Park et al., 1975, 1976), wheref
elattains a
minimum for composition waves along
[001] and is larger for waves along [100]
and [010]. Hence, the coherent spinodal
splits up into a [001], a ·101Ò, and a ·100 Ò
branch (Fig. 5-37). As a consequence the
SnO
2-rich modulations form preferentially
along [001], giving rise to a lamellar struc-
ture at later aging stages (Fig. 5-38). Forwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

416. most cubic metallic systems the elastic an-
isotropy parameter A(Sec. 5.2.4ii) is posi-
tive, rendering the minimumf
elalong the
elastically soft ·100Òdirections. According
to the CH theory the growth rate will thus
be highest along the three ·100Òdirections,
giving rise to the frequently observed mod-
ulated precipitate microstructures (Table
5-2, Figs. 5-18, 5-19b). In isotropic materi-
als such as polymers, glasses, or, for in-
stance, Fe–Cr and Fe–Cr–Co alloys, the
modulations do not grow along any prefe-
rential directions. The resulting two-phase
microstructure is of the ‘sponge-like’ type
and sometimes referred to as a ‘mottled’ or
‘interconnected’ precipitate microstructure
(e.g., Fig. 5-16, 5-19 c, 5-20). Based on
these considerations, two-phase alloys dis-
playing either lamellar,  modulatedor
interconnected precipitate microstructures
are commonly termed ‘spinodal alloys’ by
metallurgists. Furthermore, the morphol-
ogy of a two-phase material is sometimes
employed to derive the locus of the spino-
dal. This is illustrated in Fig. 5-39 for the
Cr-rich ferrite phase of a cast duplex stain-
less steel that undergoes phase separation
during tempering between 350°C and
450°C with an associated embrittlement
(Auger et al., 1989). Increasing the Cr con-
tent of the ferritic Fe–Ni–Cr solid solution
leads to the formation of Cr-rich a¢-precip-
itates during tempering. As the a¢-phase is
discernible as individual particles by
CTEM, it is concluded that phase separa-
tion occurred via nucleation and growth. A
further increase in the Cr content, to about
25 wt.%, yielded a ‘sponge-like’ micro-
structure after tempering below 400°C.
This was attributed to a spinodal mecha-
nism. Thus the spinodal is drawn as the line
that separates the two morphologies (Fig.
5-39).
It must be pointed out, however, that
interconnected or modulated structures
represent two-phase microstructures in the
later stages of the reaction (e.g., Fig. 5-18).
Even though they are widely believed to
result from spinodal decomposition in the
sense of the CH theory, their formation
might be of a rather different origin. For in-
stance, a strong elastic interaction of a high
364 5 Homogeneous Second-Phase Precipitation
Figure 5-37.The phase 
diagram and the spinodal
curves for composition
waves along [001], ·101Ò,
and  ·100Òdirections for
elastically anisotropic te-
tragonal TiO
2–SnO
2. The
spinodals were calculated
on the basis of the regular
solution model. (From
Park et al., 1976.)www.iran-mavad.com 
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417. 5.5 Early Stage Decomposition Kinetics 365
number density of individual nuclei, each
of which is surrounded by a solute-de-
pleted zone in which no further nucleation
can occur, may also lead to regularly ar-
ranged precipitates, i.e., modulated struc-
tures (Ardell et al., 1966; Doi et al., 1984,
1988; Doi and Miyazaki, 1986). On the
other hand, initially interconnected micro-
structures in Cu–Ni–Fe alloys were found
to break up into isolated plates (Piller et al.,
1984). The later-stage microstructure is
thus neither sufficient to draw any conclu-
sions on the early-stage decomposition
mode nor for a definition of a ‘spinodal
alloy’. We are therefore still left with the
question as to what really is a spinodal al-
loy? To answer this question unequiv-
ocally, we must verify by any microanalyt-
ical technique that the amplitude of the
composition waves increases gradually
with time until the evolving second phase
has finally reached its equilibrium compo-
sition. Such an experimental verification is
a difficult task and, to our knowledge, was
only shown for decomposing Fe–Co–Cr 
(Figs. 5-16 and 5-17), AlNiCo permanent
magnetic materials (Hütten and Haasen,
1986) and Fe–Cr (Brenner et al., 1984) by
means of AFIM, and for a phase separating
polystyrene–polyvinylmethylether poly-
mer mixture employing nuclear magnetic
resonance methods (Nishi et al., 1975).
As has ben outlined in Sec. 5.2.3, re-
gardless of whether phase separation oc-
cours via nucleation and growth or via spi-
nodal decomposition, the underlying mi-
croscopic mechanism is diffusion of the
solvent and solute atoms. In this sense
there is thus no need to distinguish between
the two different decomposition modes and
the term ‘spinodal alloy’ is simply seman-
tic in nature. This is also reflected in the
various attempts to develop ‘unified
theories’ comprising either mode.
From the practical point of view the mi-
crostructure of virtually all technical two-
phase alloys corresponds to that of the later
stages. Hence, the practical metallurgist
worries little about the initial stages of un-
mixing but is instead interested in predicting
the growth and coarsening behavior of pre-
cipitate microstructures in the later stages.
This will be the subject of Secs. 5-6 and 5-7.
Figure 5-38.CTEM micrograph of equimolar
TiO
2–SnO
2displaying a lamellar structure consist-
ing of alternating TiO
2- and SnO
2-rich layers formed
after aging at 900 °C for 60 min. From Park et al.
(1976).
Figure 5-39.Variation of the phase boundaries in
the Fe–Ni–Cr ferrite phase with chromium content as derived from microstructural observations. (After Auger et al., 1989.)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

418. 366 5 Homogeneous Second-Phase Precipitation
Table 5-2.TEM and AFIM studies on phase separating solids that revealed ‘spinodal precipitate
microstructures’. If not otherwise stated the concentrations are in at.%.
Decomposing solid Experimental Morphology of two- Authors
technique phase microstructure
Metallic alloys
Al–4 wt.% Cu TEM (‘side-bands’) Modulated structure Rioja and Laughlin (1977)
Al–(2.4–3%) Li HREM, Radmilovic et al. (1989)
X-ray diffraction
Cu
3–4Mn
xAl TEM (‘side-bands’) Modulated structure; Bouchard and Thomas (1975)
at later stages
Cu
2MnAl + Cu
3Al-plates
Cu–(1.5–5.2 wt.%) Ti TEM (‘satellite Modulated structure Laughlin and Cahn (1975)
analysis’)
Cu–2.7% Ti AFIM Modulated structure Biehl and Wagner (1982)
Ni–12% Ti AFIM Modulated structure Grüne (1988)
Ni-29% Cu–21% Pd TEM Modulated structure Murata and Iwama (1981)
Ni-base superalloys AFIM Short wavelength Bouchon et al. (1990)
(≈2.5 nm) Cr fluctuation
in 
g-phase
Nimonic 80 A TEM (‘side-bands’) Modulated structure Wood et al. (1979)
Fe–25% Be AFIM Modulated structure Miller et al. (1984, 1986)
CuNiFe AFIM Interconnected percolated  Piller et al. (1984)
structure (cf. Fig. 5-19c)
CuNiFe TEM (‘side-bands’) Modulated structure Wahi and Stager (1984)
(cf. Fig. 5-18) Livak and Thomas (1974)
Co–10% Ti FIM Modulated structure Davies and Ralph (1972)
Co–3 wt.% Ti TEM (‘side-bands’) Modulated structure Singh et al. (1980)
–1 … 2 wt.% Fe
Mn–30 wt.% Cu  TEM, magnetic Interconnected Yin et al. (2000)
susceptibility structure
Steels
Cast Duplex stainless TEM, AFIM After decomposition  Auger et al. (1989)
original ferritic
Fe–Cr–Ni shows
‘sponge-like’ structure
Ferrous martensite
Fe–15 wt.% Ni–1 wt.% C TEM Tweed structure Taylor (1985)
Fe-25 wt.% Ni TEM Tweed structure Taylor (1985)
–0.4 wt.% C
Fe–13 … 20 wt.% Mn X-ray diffraction Modulated structure at  Miyazaki et al. (1982)
(‘side-bands’), TEM early times; isolated
Fe
2Mo particles at later
stages
Amorphous alloys
Ti
50Be
40Zr
10 AFIM Wavy composition profile Grüne et al. (1985)www.iran-mavad.com
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419. 5.5 Early Stage Decomposition Kinetics 367
5.5.6 Monte Carlo Studies
The basic features of early stage decom-
position as a stochastic process, without ar-
tificial distinction between nucleation,
growth and coarsening regimes, have been
extensively studied in model alloys by
means of Monte Carlo (MC) computer sim-
ulations, mainly by Lebowitz, Kalos and
coworkers (Lebowitz and Kalos, 1976;
Binder et al., 1979; Marro et al., 1975,
1977; Penrose et al., 1978; Lebowitz et al.,
1982; see also Binder and Fratzl, 2001;
Chapter 6 of this volume). The binary model
alloys are usually described in terms of a
three-dimensional Ising model with pairwise
nearest-neighbor interactions on a simple
rigid cubic lattice the sites of which are oc-
cupied by either A or B atoms, and a phase
diagram which displays a symmetrical mis-
cibility gap centered at 50 at.% solute con-
centration (Fig. 5-40). The microscopic dy-
namics of this system have commonly been
described by the Kawasaki model (Kawa-
saki, 1972). There a nearest-neighbor pair
of lattice sites is chosen at random, then the
atoms on those sites may be interchanged
with a probability that depends on the ener-
gies of the configuration before and after
the exchange in such a way that detailed
balancing holds (Penrose, 1978). Monte
Carlo techniques are then employed to
carry out this stochastic process.
The Kawasaki dynamics employed for
MC simulations are far from being repre-
sentative of a real binary alloy system as
Table 5-2.Cont.
Decomposing solid Experimental Morphology of two- Authors
technique phase microstructure
Glasses, ocides
TiO
2–60 mole% SnO
2 TEM Tetragonal system with  Stubican and Schultz (1970)
lamellar modulations
along [001]
TiO
2–50 mole% SnO
2 HREM Tetragonal system with  Horiuchi et al. (1984)
lamellar modulations
along [001]
SiC–(50–75 mole%) AlN TEM (‘satellites’) Modulated structure Kuo and Virkar (1987)
Figure 5-40.Phase diagram of the 3-dimensional Is-
ing model approximating a binary model alloy. In
terms of the mean field theory (Sec. 5.2.2) ‘alloys’
#1 to #4 are quenched into the metastable regime, 
‘alloys’ #5 to # 7 beyond the classical spinodal line
are quenched into the unstable region of the phase di-
agram. (After Lebowitz et al., 1982.)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

420. the atomic exchange is assumed to occur
directly rather than indirectly via the va-
cancy mechanism. Nevertheless, MC simu-
lations based on Kawasaki dynamics on an
atomic scale yielded the time evolution of
the cluster configuration and the structure
function S(
k, t) of model alloys quenched
into any region below the solubility line
(Fig. 5-40) without worrying about com-
plicating factors, such as an insufficient
quenching rate, excess vacancies, or lattice
defects, which we commonly face when in-
vestigating real alloys (Sec. 5.3.2). Fur-
thermore, MC simulations have allowed a
critical examination of the various theoret-
ical approaches in terms of cluster dynam-
ics models or spinodal models to be made.
For practical computational limitations,
however, the maximum size of the model
alloy has commonly been restricted to
about 50¥50¥50 lattice sites. Because of
the size limitation, in general MC kinetic
experiments can only cover the earlier
stages of a precipitation reaction in alloys
where the supersaturation is sufficiently
high for the formation of a large number
density of clusters or nuclei, and where the
cluster sizes are still considerably smaller
than the linear dimension of the model
system. This is frequently not the case in
real alloys.
In essence, the results from MC simula-
tions based on Kawasaki dynamics have re-
covered the predictions both from the gen-
eralized nucleation theory of Binder and
coworkers (Sec. 5.5.3) and from the non-
linear spinodal LBM approach. In particu-
lar, the time evolution of the structure func-
tion in the kinetic Ising model displays
qualitatively the same features as the corre-
sponding ones obtained from the latter the-
ories (cf. Figs. 5-31 and 5-35b) or from ex-
periment (e.g., Fig. 5.32a).
For instance, S(k,t) of ‘alloy’ #4 (Fig.
5-41b), which according to Fig. 5-40 lies
close to the spinodal line within the meta-
stable regime, evolves similarly to that of
‘alloy’ #5 quenched into the center of the
spinodal region (Fig. 5-41c). Thus, in
agreement with the generalized nucleation
theory and the LBM spinodal approach, but
unlike the predictions of linearized CH the-
ory of spinodal decomposition, MC simu-
lations again reveal i) noevidence for any
abrupt change of the decomposition kinet-
ics on crossing the spinodal curve, ii) no
common cross-overpoint of the S(k)
curves taken after different aging times, iii)
no exponential growth of the scattering in-
tensity for a certain time-independent wave
vector in any time regime, and, further-
more, iv) the peak position of S(
k,t) at k
m
is notfound to be time-independent but is
shifted towards smaller values of 
kindicat-
ing the immediate growth of clusters.
Frequently, the MC data for the time ev-
olution of the peak position (
k
m) and the
peak height (S
m) of the structure function
have been fitted to simple power laws (cf.
Sec. 5.8.2), such as 
k
m(t)µt
–a
and S
m(t)
dt
b
; aand bwere estimated to range from
0.16 to 0.25 and 0.41 to 0.74, respectively,
depending on the initial supersaturation of
the ‘alloy’ (Marro et al., 1975, 1977; Sur et
al., 1977). Lebowitz et al. (1982) pointed
out, however, that due to the finite (small)
size of the system it is difficult to extract
from computer simulations precise and re-
liable information about the analytical
form of 
k
m(t), for example and that it is
possible to fit the same MC data with other
functional forms than the power laws given
above. (In Sec. 5.8.2 we will show, in fact,
that apart from the late stages of coarsen-
ing, it is usually not feasible to interpret ex-
perimental kinetic data over an extended
aging period in terms of a power-law be-
havior with a time-independent exponent.)
The Kawasaki dynamics are based on an
unrealistic exchange mechanism between
368 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

421. 5.5 Early Stage Decomposition Kinetics 369
neighboring atoms. In order to account for
the more realistic case of atomic diffusion
being based on the vacancy mechanism,
the Ising model was more recently ex-
tended to binary alloys comprising vacan-
cies (Yaldram and Binder, 1991 a,b; Fratzl
and Penrose, 1994). Comparing the growth
rates of clusters in the Ising model of a
two-dimensional model alloy showed that
the asymptotic regime of cluster coarsen-
ing (R
–
~t
1/3
; cf. Sec. 5.6.2) is approached
much faster with vacancy dynamics than
Figure 5-41.Time evolution of the structure func-
tion at T/T
c= 0.59 as obtained from MC computer
simulations. With reference to Fig. 5-40:
a) for ‘alloy’ #1 with c
0= 0.05 (after Lebowitz et al.,
1982),
b) for ‘alloy’ #4 with c
0= 0.2 (after Sur et al., 1977)
and
c) for ‘alloy’ #5 with c
0= 0.5 (after Marro et al.,
1975). The given times are in units of a Monte Carlo
step, i.e., the average time interval between two at-
tempts at exchanging the occupancy of a specific
site. The numerical results for S(
k,t) at the discrete
values of 
kwere connected by straight lines.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

422. with Kawasaki dynamics, in particular at
larger supersaturations; the cluster shapes
are found not to depend on the particular
dynamics model (Fratzl and Penrose,
1994).
In a first attempt to interpret experimen-
tal data with kinetic data obtained from
MC simulations, Soisson et al. (1996) sim-
ulated phase separation in a Fe–1.34 at.%
Cu alloy. The alloy was modeled as a rigid
b.c.c. crystal with 2 ¥L
3
lattice sites (Lvar-
ying from 32 to 64) containing one single
vacancy; the energetics and the parameters
of the atomistic kinetic model were ad-
justed to the thermodynamic and diffusion
data of the Fe–Cu system. The MC results
confirmed earlier results from magnetic
SANS studies (Beaven et al., 1986) that the
precipitating clusters are pure copper; fur-
thermore the time evolution of the precipi-
tated volume fraction agreed quite well
with the one experimentally determined
from resistivity measurements (Lê et al.,
1992). In the later stages dynamical scaling
behavior was shown to hold, indicating that
the cluster pattern remains similar as aging
time increases (cf. Sec. 5.8.1).
5.6 Coarsening of Precipitates
5.6.1 General Remarks
For most two-phase alloys, the simple
model of diffusional growth of isolated
non-interacting particles with uniform size,
on which Eq. (5-47) is based, frequently
does not give a realistic description of the
further dynamic evolution of the precipi-
tate microstructure beyond its nucleation
stage. In reality, towards the end of the nu-
cleation period a more or less broad parti-
cle size distributionf(R) is established
(Fig. 5-10 and 5-29). According to the
Gibbs–Thomson equation (5-45), the solu-
bility c
R(R) in the presence of small parti-
cles with a large ratio of surface area to
volume is larger than that for largerones.
With reference to Eq. (5-43), this leads to a
size-dependent growth rate, which is posi-
tive for larger particles with c ¯>c
Rand neg-
ative for smaller ones with c¯<c
R. The
growth rate becomes zero for particles with
c¯=c
Rwhich are in unstable equilibrium
with the matrix. Their radius R* is derived
from Eq. (5-46) as
(5-61)
Hence, driven by the release of excess
interfacial energy, larger precipitates will
grow at the expense of smaller ones which
dissolve again given rise to a change in the
precipitate size distribution. This process,
which is commonly referred to as coarsen-
ing orOstwald ripening
2
, frequently re-
duces the precipitate number density form
≈10
25
m
–3
to less than 10
19
m
–3
in typical
two-phase alloys during aging (cf. Fig.
5-3). Usually, the coarsening process is
considered to be confined to the latest
stages of a precipitation reaction. However,
as will be shown in Sec. 5-7, coarsening
may accompany the growth process out-
lined in Sec. 5.5.2, or may even start while
the system is still in its nucleation period,
depending on the initial supersaturation of
the solid solution.
5.6.2 The LSW Theory of Coarsening
In essence, the coarsening of randomly
dispersed second-phase particles is a multi-
particle diffusion problem which is diffi-
cult to handle theoretically. In their classic
LSW coarsening theory, Lifshitz and Slyo-
R
V
RT cc
K
cc
*
ln ( / ) ln ( / )
≈≡ ′
2 11
s
s
ab b
a
ab
ag
ee
370 5 Homogeneous Second-Phase Precipitation
2
In its original meaning, Ostwald ripening is con-
fined to a coarsening reaction where the second-
phase particles act as the only sinks or sources of so-
lute atoms.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

423. 5.6 Coarsening of Precipitates 371
zov (1961), and Wagner (1961) calculated
the time evolution off(R, t) which satis-
fies the continuity equation:
(5-62)
On the basis of the continuity equation and
of Eq. (5-43), the time evolution of the
mean particle radius R
–
(t) and the precipi-
tate number density N
v(t) are derived. Cer-
tain approximations, however, had to be
made in order to solve the equations of mo-
tion analytically: 
a) Both terminal phases aand bare dilute
solutions; their thermodynamics can be
described by a dilute solution and the
linearized version (Eq. 5-46) of the
Gibbs–Thomson equation may be used. 
b) The precipitated volume fraction f
p=
(4
p/3) R
–
3
N
vis close to zero. In such a
dilute system interparticle diffusional
interactions such as those occurring in
more concentrated alloys (cf. Sec.
5.6.3) can be neglected, and a particle
only interacts with the infinite matrix.
c)f
p≈const., i.e., the decomposition is
close to completion with the supersatu-
ration Dc≈0. This inherently confines
the LSW theory to the late stages of a
precipitation reaction.
With these assumptions the LSW theory
yields in the asymptotic limit Dc(t)Æ0
temporal power laws
R
–
3
(t)=K
R
LSWt (5-63a)
N
v(t)= K
N
LSWt
–1
(5-63b)
Dc(t) = K
C
LSWt
–1/3
(5-63c)
for the time evolution of the average parti-
cle radius, the particle density and the
supersaturation, respectively, and a particle
size distribution f
LSW(R/R
–
), the shape of
which is time invariant under the scaling of
the average particle size R
–
(Fig. 5-42). It
∂
∂
+
∂
∂
⎡
⎣
⎢
⎤
⎦
⎥
=
f
tR
f
R
t
d
d
0
must be emphasized that these predictions
hold only in the limit tapproaching infin-
ity, since the particle size distribution that
is present at the beginning of coarsening
can be quite different from the time-invari-
ant form.
Extensions of the LSW theory to binary
systems with non-zero solubilities and non-
ideal solution thermodynamics, and ade-
quate modification of the Gibbs–Thomson
equation, have reproduced the temporal
power laws (Eq. 5-63). The corresponding
rate constants for this more realistic case
were derived as (Schmitz and Haasen,
1992; Calderon et al., 1994):
(5-64a)
(5-64b)
(5-64c)
K
cc
D
K
C
ee
R=
9
4
23
(–)
/ba
K
f
Kf
N
p
R=
3
4
1
3p
K
DV
cc G
R ee=
′′
8
9
2
babb
s
aa(–)
Figure 5-42.CTEM analyses of the normalized  g¢-
particle size distribution in Ni–8.74 wt.% Ti after ag-
ing at 692°C for the given times. For comparison, the
shape-invariant distribution function f
LSWof the
LSW theory is included. (After Ardell, 1970.)www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

424. G
a≤ is the second derivative of the molar
Helmholtz energy of the aphase; f
3is the
third moment of the time-independent
scaled particle radius distribution function 
f(R/R
–
) which in the zero volume fraction
limit (i.e. f= f
LSW; see Sec. 5.6.3) is 1.129.
For an ideal solution and the dilute solu-
tion limit (i.e. c
a
e∫1, c
b
e≈1) the rate con-
stant K
R(Eq. 5-64a) adopts the form
(5-65)
of the original LSW theory (Calderon et
al., 1994).
Numerous experimental studies on a
wide variety of two-phase alloys attempted
to examine the coarsening kinetics pre-
dicted by the LSW theory. Regardless of
the particular alloy system and the micro-
analytical technique employed (e.g.,
CTEM on Ni–Al, Ni–Ti: Ardell, 1967,
1968, 1970; and on Ni–Si: Cho and Ardell,
1997; AFIM on Ni–Al: Wendt and Haasen,
1983; SANS on Fe–Cu: Kampmann and
Wagner, 1986), these studies frequently re-
vealed the experimental size distribution
function to be considerably broader than
f
LSW(R/R
–
)(Fig. 5-42), whereas a plot of R
–
3
or N
v
–1vs. tyielded more or less straight
lines (due to limited statistics, the error
bars are usually rather large). From the
slopes of these LWS plotsthe product 
s
abD
can be derived with c
a
eusually taken from
the known phase diagram. Frequently Dc
was also measured and plotted versus t
–1/3
(Eq. (5-63c)) in order to determine D/ s
2
ab
(e.g., Ardell, 1967,  1968, 1995; Wendt 
and Haasen, 1983). Thus apparently abso-
lute values for both 
s
aband Dhave been
determined from values of 
s
abD and
D/
s
2
ab
. However, the rate constants K
R
LSW
and K
C
LSWin Eq. (5-63) were derived 
with the assumption c): Dc≈0 or f
p= const.
Once this condition is fulfilled, it is no
K
DV c
RT
R
LSW
e
g=
8
9
baab
s
longer feasible to follow minor changes of Dcwith time quantitatively by any of the
experimental techniques frequently em- ployed, such as CTEM, AFIM, SAXS or SANS (Sec. 5.3.1); in the asymptotic limit Dc
Æ0, it even appears difficult to measure
Dc(t) in alloys containing a ferromagnetic
phase with magnetic techniques (Ardell, 1967, 1968), though these direct methods are certainly more sensitive than CTEM, AFIM or SAS. On the other hand, in those earlier decomposition stages, where Dc(t)
is experimentally accessible, an LSW anal- ysis cannotbe performed, and, in particu-
lar, the rate constant K
R
LSW(Eq. (5-65)) is
no longer valid (see Sec. 5.7.4.3). Thus, in practice, an LSW analysis based on Eqs. (5-63) and (5-65), at its best, can be per- formed only in dilute systems and yields only a value for the product D
s
ab. This fact
was ignored in most LSW analyses based on the independent measurement of both R
–
(t
1/3
) and Dc (t
–1/3
); hence, the values of
Dand 
s
abderived from this type of analy-
sis must be regarded with some reserva- tion. However, frequently the interfacial energy was derived exclusively from Eq. (5-65) under the assumption that the effec- tive diffusivity Dis identical with that ob-
tained from an extrapolation of available high-temperature data to the aging temper- ature, thus neglecting the influence of quenched-in vacancies (Sec. 5.3.2.1). (In Sec. 5.7.4.4 two different methods will be discussed which allow a separate deriva- tion of 
s
aband Dto be made from experi-
mental data, one method without even knowing Dc(t).)
For non-ideal solid solutions, 
s
abmust
be derived from Eq. (5-64a). Apart from knowing D, this requires detailed informa-
tion on the thermodynamics of the amatrix
phase in order to derive G
a≤. For many bi-
nary alloys the necessary thermodynamic functions have become available via the
372 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

425. 5.6 Coarsening of Precipitates 373
CALPHAD method (Sec. 5.2.4). If not
available, they must be derived from more
or less adequate solution models, with the
regular solution model being the less so-
phisticated one. For the case of a binary
model alloy with a miscibility gap and reg-
ular solution behavior, Calderon et al.
(1994) demonstrated that depending on ag-
ing temperature, 
s
abmay be significantly
different if assessed from the correct form
of the rate constant for a regular solution 
or from the original LSW form, i.e. Eq. 
(5-65).
5.6.3 Extensions of the Coarsening
Theory to Finite Precipitate Volume
Fractions
There have been many efforts to extend
the LSW theory to the more realistic case
of a finite precipitated volume fraction and
to investigate its influence on the shape of
the distribution function and the coarsen-
ing kinetics (Ardell, 1972). The central
challenge is to determine the effects of
interparticle diffusional interactions on the
growth rate of a particle of a given size.
Theories accounting for these diffusional
interactions fall into two broad categories:
effective medium theories, and statistical
theories that are based upon a solution to
the diffusion field in this multiparticle
system. All theories are in agreement that
the presence of a non-zero volume fraction
does not change the exponents of the tem-
poral power laws given by LSW (Eq. 
(5-63)) or the existence of a time-indepen-
dent scaled particle size distribution. The
presence of a non-zero volume fraction,
however, does change the amplitudes of the
temporal power laws and the shape of the
scaled distribution functions. The rate con-
stant K
R
LSWincreases with the volume frac-
tion (Fig. 5-43) and the time-independent
distribution function becomes broader and
more symmetric as the volume fraction in-
creases (Fig. 5-44). The reason for the in-
crease in the rate constant is clear: as the
volume fraction increases, shrinking parti-
cles move closer to growing particles and
thus the concentration gradients are larger,
and the rate of growth and shrinkage in-
creases.
The effective medium theories deter-
mine the growth or shrinkage rate of a par-
ticle of a given size using a medium that is
constructed, presumably, to give the statis-
tically averaged growth rate of particles of
a certain size. Examples of such theories
are due to Tsumuraya and Miyata (1983),
Brailsford and Wynblatt (1979) and, more
recently, Marsh and Glicksman (1996).
While the effective medium employed by
Brailsford and Wynblatt has been shown
analytically to be consistent with a solution
to the multiparticle diffusion equation, this
Figure 5-43.The rate constant K, relative to that of
LSW, K
LSW, as a function of the volume fractionf
P.
Shown on the figure are the predictions of Brailsford
and Wynblatt (BW), Marsh and Glicksman (MG),
Marqusee and Ross (MR), Tokoyama and Kawasaki
(TK) and the simulations of Akaiwa and Voorhees
(AV) where 
∫denote the use of the monopole ap-
proximation of the diffusion field and 
≤denote
monopole and dipole approximations to the diffusion
field.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

426. has not been done with the other theories.
This, then, is the major disadvantage of
these theories. A significant advantage,
however, is that they are simple to apply
and make predictions over a wide range of
volume fractions, even as high as 0.95. In
addition, both the Brailsford and Wynblatt
and Marsh and Glicksman theories corre-
spond reasonably well with the first-princi-
ples statistical theories for volume frac-
tions below 0.3. Above 0.3, these are the
only theories that make predictions on the
rate constants and particle size distribu-
tions.
The statistical theories are based upon a
solution to the multiparticle diffusion prob-
lem. The particles are assumed to be spher-
ical and the concentration in the matrix at
the interface of each particle is given by the
Gibbs–Thompson equation, (5-45). For
volume fractions below 0.1 the solution to
the diffusion equation is represented as a
monopole source or sink of solute in the
center of each particle (Weins and Cahn,
1973). At higher volume fractions dipolar
terms must be included (Akaiwa and Voor-
hees, 1994). Given, then, a spatial distribu-
tion of particles and a particle size distribu-
tion, it is possible to determine the coarsen-
ing rate of each particle. The more chal-
lenging step is to determine from this infor-
mation the statistically averaged growth
rate of a particle of a given size. This can
be done analytically (Marqusee and Ross,
1984; Tokuyama and Kawasaki, 1984; Yao
et al., 1993), or numerically by placing a
large number of particles in a box and de-
termining their coarsening rate (Voorhees
and Glicksman, 1984; Akaiwa and Voo-
rhees, 1994; Mandyam et al., 1998). Al-
though all of these theories begin with the
same solution to the diffusion equation,
none of the predictions for the rate con-
stants and particle size distributions are in
agreement, illustrating the difficulty in per-
forming the statistical averaging.
Nevertheless, a number of qualitative ef-
fects of a finite volume fraction become
clear after examining the predictions of
these theories. (a) The growth rate of a par-
ticle of a given size is a function of its sur-
rounding particles. For example, if a parti-
cle of size R
1is surrounded by particles of
radii R<R
1, this particle will grow, but if
this particle is surrounded by particles with
radii R>R
1, it will shrink. Thus, unlike in
the LSW theory the growth rate of a parti-
cle is not solely a function of its size. (b)
The local diffusional interactions give rise
to spatial correlations between particles
that are not random. For example, the aver-
age interparticle separation for a system
undergoing coarsening is larger than that
for a random spatial distribution. This is
because the probability of finding a small
and large particle almost touching, which
can occur in a system with a random spatial
distribution, is low, as the strong diffu-
sional interactions that occur when a large
and a small particle are located close to
374 5 Homogeneous Second-Phase Precipitation
Figure 5-44.Steady-state (time-invariant) precipi-
tate size distributions at various volume fractions.
For comparison the corresponding LSW distribution
for zero volume fraction is also shown. (After To-
kuyama and Kawasaki, 1984.)www.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

427. 5.6 Coarsening of Precipitates 375
each other causes this small particle to dis-
appear. At larger interparticle separation
distances, however, large particles tend to
be surrounded by small particles as they
are feeding solute to the growing large par-
ticles (Akaiwa and Voorhees, 1994). Thus,
there are both spatial and particle size cor-
relations (Marder, 1987). Information on
the spatial correlations between particles is
needed to determine the structure function
that is measured using small angle scatter-
ing. The agreement between the predic-
tions of Akaiwa and Voorhees (1994) for
the structure function and the experimen-
tally measured structure function in Al–Li
alloys is reasonable (Che et al., 1997). (c)
Another effect of interparticle diffusional
interactions is that the center of mass of the
particles is not fixed, but moves in a man-
ner consistent with the concentration gra-
dients in the system. This particle motion
has been determined theoretically (Mar-
der,1987; Akaiwa and Voorhees, 1994), and
has been observed in experiments on coars-
ening in transparent solid–liquid mixtures
(Voorhees and Schaefer, 1987).
Experiments in which the coarsening ki-
netics of solid particles in a liquid (Fig. 
5-45) have been measured show clearly
that the coarsening rate increases with vol-
ume fraction (Hardy and Voorhees, 1988,
and references therein). In two-phase solid
systems, however, there are reports that the
rate constant is independent of the volume
fraction (Cho and Ardell, 1997). The rea-
son for these contradictory results may be
the elastic stress that is present in the two-
phase solid systems. Although the pre-
dicted particle size distributions of the non-
zero volume fraction theories are broader
than those of LSW, nearly all experimen-
tally measured particle size distributions
are broader than the predictions of the non-
zero volume fraction theories. Until re-
cently there have been no experiments per-
formed in a system wherein the materials
parameters, such as the interfacial energy
and diffusion coefficient, are known (inde-
pendent of a coarsening experiment) and
where this system satisfies all the assump-
tions of theory. Therefore, it is unclear if
the disagreement between theory and ex-
periments is due to an artifact of the system
employed in the experiments or a defect in
the theories. Recent experiments that em-
ploy a system in which the materials pa-
rameters are known and satisfy all the as-
sumptions of theory have been performed
using solid Sn-rich particles in a Pb-rich
eutectic liquid (Alkemper et al., 1999).
There was no convective motion of the par-
ticles because the experiments were per-
formed in the microgravity environment of
the Space Shuttle. These experiments show
that there can be very long transients asso-
ciated with the coarsening process. In par-
ticular, during the course of the experiment
the initially broad particle size distribution
evolved slowly towards that predicted by
theory. Even with a factor of four change in
the average particle size the distribution
never reached the steady-state distribution
predicted by theory. This may be the reason
why the experimentally measured particle
size distributions (measured using other
systems) rarely agree with theory. Never-
theless, the evolution of the distribution
was found to correspond quite well with
the predictions of two theories for transient
Ostwald ripening.
Coarsening of two-phase alloys with
modulatedmicrostructures consisting of
large volume fractions (f
p≥30%) of iso-
latedcoherently misfitting particles, e.g.,
Cu–Ni–Fe and Cu–Ni–Cr (Fig. 5.19),
Cu–Ni–Si (Yoshida et al., 1987), or mod-
ulated Co–Cu (Miyazaki et al., 1986), was
reported to be rather sluggish. The time ex-
ponent aof the coarsening rate, defined by
R
–
µt
a
,of these alloys was consistentlywww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

428. 376 5 Homogeneous Second-Phase Precipitation
Figure 5-45.Evolution of Sn-rich particles withf
p= 0.64 in a liquid Pb–Sn matrix during coarsening at 185°C.
The top row is at constant magnification and illustrates a typical coarsening process. For scaling the absolute
size of the microstructure, in the bottom row the magnification has been multiplied by the ratio of the average
intercept length L
–
at time tto L
–
at t= 75 min. (From Hardy and Voorhees, 1988.)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

429. 5.6 Coarsening of Precipitates 377
found to be less than 1/3, the value ex-
pected from conventional coarsening the-
ory on the basis of Ostwald ripening. Fur-
thermore, it was shown that coarsening of a
·100Òmodulated structure in less concen-
tratedCo–Cu alloys (≤20 at.% Cu;f
p≈
20%) still follows the t
1/3
kinetics, whereas
the coarsening rate becomes extremely
small in a Co–50 at.% Cu alloy (f
p≈50%),
as indicated by the small time exponent 
a< 1/50 (Miyazaki et al., 1986).
As was pointed out in Sec. 5.5.5 “mot-
tled structures” or “interconnected micro-
structures” can result from spinodal de-
composition of alloys with vanishingcohe-
rency strains. Unlike for modulated struc-
tures, elastic interactions can therefore be
ruled out as a likely reason for the stability
of interconnected microstructures against
coarsening. Nevertheless, the coarsening
rate of interconnected phases is frequently
also foundnotto follow the t
1/3
kinetics.
Experimental studies on the coarsening ki-
netics of interconnected phases in hard
magnet materials such as Fe–Cr–Co (Zhu
et al., 1986), Al–Ni–Co (Hütten and Haa-
sen, 1986; Katano and Iizumi, 1982) in-
stead yielded time exponents abetween 1/4
and 1/10. Evidently, an adequate theoreti-
cal description of the coarsening kinetics of
both modulated and mottled microstruc-
tures has to go beyond the mere modeling
of diffusional interaction between an en-
semble of isolated sphericalparticles.
5.6.4 Other Approaches Towards
Coarsening
In addition to the LSW-type coarsening
mechanism, which is based on the evopora-
tion and condensation of single atoms from
dissolving and growing precipitates,
Binder and Heermann (1985) also con-
siderred a cluster-diffusion-coagulation
mechanismlikely to become operative dur-
ing the intermediate stages of coarsening.
Depending on the specific (local) micro-
scopic diffusional mechanism which is as-
sumed to contribute to the shift of the cen-
ter of gravity of the particles and their
likely coagulation, the time exponents afor
the related coarsening rate were evaluated
between a= 1/6 and a= 1/4 and hence are
smaller than predicted by the LSW theory
(a= 1/3).
According to Fig. 5-32b the position of
the SANS peak intensity at 
k
mvaries in
proportion to t
–0.25
. This might, in fact, be
interpreted in terms of a cluster-diffusion-
coagulation mechanism being dominant
prior to LSW-type coarsening. In Sec.
5.8.2 it will be shown, however, that even
if single atom evaporation or condensation
in the LSW sense is assumed to be the only
microscopic mechanism contributing to
particle growth, the time exponent aal-
ready displays a strong time dependence.
Depending on the initial supersaturation of
the alloy, a(t) may then vary between al-
most zero and 0.5 during the course of a pre-
cipitation reaction. The question of whether
aeven reaches its asymptotic value 1/3 de-
pends on whether the experiment spans a
sufficiently long range of aging times.
From the experimental point of view it
thus does not appear feasible to decide
merely on grounds of the measured time
exponent whether the cluster-diffusion-
coagulation mechanism influences the
growth or coarsening kinetics of a precipi-
tate microstructure.
5.6.5 Influence of Coherency Strains
on the Mechanism and Kinetics
of Coarsening – Particle Splitting
The LSW theory, as well as the exten-
sions to finite volume fractions, assume
that the coarsening process is driven en-
tirely by the associated release of interfa-www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

430. cial energy. In the presence of elastic
stresses, however, such as those induced by
coherency strains, the coarsening process
can be driven by the release of both the
interfacial and elastic energy. The elastic
energy can be decomposed into an elastic
self-energy, the elastic energy of an iso-
lated particle in an infinite medium, and an
interaction energy that is due to the pres-
ence of other particles in the system. The
elastic self-energy is a strong function of
the shape of a particle and is responsible
for the shape bifurcations mentioned in
Sec. 5.4.2. Moreover, the elastic interac-
tion energy is also a function of the shape
of the particles, the importance of which
has been emphasized by Onuki and Nishi-
mori (1991).
In the presence of misfit strains, inter-
particle elastic interactions can give rise to
pronounced spatial correlations between
the precipitates. Ardell et al. (1966) pro-
posed that the elastic interactions between
g¢–Ni
3Al particles are responsible for the
alignment of the particles along the elasti-
cally soft directions of the crystal during
coarsening. The resulting structure of
many nickel-based alloys reveals that the
378 5 Homogeneous Second-Phase Precipitation
Figure 5-46.High-magnification scanning
electron micrograph of concave 
g¢-precipi-
tates in an aged Ni–23.4 Co–4.7Cr–4 Al
– 4.3Ti (wt.%) superalloy prior to splitting.
(Reproduced by courtesy of D. Y. Yoon
(Yoo et al., 1995).)
Figure 5-47.As in Fig. 5-46, showing a
split 
g¢-precipitate.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

431. 5.6 Coarsening of Precipitates 379
g¢-type precipitates are rather uniformly
distributed (Fig. 5-19 b), which is some-
times reminiscent of modulated structures.
These strong spatial correlations develop
by two basic mechanisms. The first is that
particles aligned as stringers along the
elastically soft directions will grow at the
expense of particles not so aligned (Ardell
et al., 1966). The second is that alignment
will occur by particles migrating through
the matrix (Voorhees and Johnson, 1988).
This migration is a result of the elastic
interactions inducing a non-uniform inter-
facial concentration and, hence, non-uni-
form concentration gradients along the par-
ticle interface.
It has been observed (Miyazaki et al.,
1982; Doi et al., 1984; Kaufmann et al.,
1989) that during coarsening, individual
g¢-type precipitates can split into groups of
two or eight smaller cuboidal particles. Us-
ing a deep etching technique and a scan-
ning electron microscope, the recent experi-
ments of Yoo et al.(1995) have illustrated
clearly the three-dimensional morphology
of the particles undergoing the splitting
process (Figs. 5-46 and 5-47). Such a split-
ting process clearly cannot be driven by a
reduction in interfacial energy and thus the
cause of the splitting has been ascribed to a
decrease in the elastic energy. Although the
elastic self-energy does not change on
splitting, assuming that the particle mor-
phologies are the same before and after
splitting,if the elastic interaction energy is
negative, e.g., as is the case for particles
aligned along the elastically soft directions
of an elastically anisotropic crystal, the to-
tal elastic energy will decrease upon split-
ting (Khachaturyan and Airapetyan, 1974;
Khachaturyan et al., 1988; Johnson and
Cahn, 1984; Miyazaki et al., 1982). How-
ever, constraining the morphology of the
particles to be invariant upon splitting ne-
glects the effects of the elastic interaction
on the morphology of the particles and
hence on the magnitude of the elastic inter-
action energy itself. In fact, two-dimen-
sional calculations (Thompson et al., 1994)
wherein the morphology of a particle was
not constrained found that a misfitting
fourfold symmetric particle in an elasti-
cally anisotropic homogeneous crystal was
stable with respect to interfacial perturba-
tions at least up to L= 26 (cf. Sec. 5.4.2).
Thus the conclusion reached on the basis of
elastic energy considerations that splitting
is possible appears to be due to the assump-
tion that the morphology of the precipitate
is invariant upon splitting. Miyazaki et al.
(1982) proposed that the splitting process
begins with the formation of the matrix
phase in the middle of the precipitate. This
idea was verified through the diffuse inter-
face calculations of Wang et al. (1991).
However, the experimental results of Yoo
et al. (1995) confirmed the hypothesis of
Kaufmann et al. (1989) that the splitting
process is initiated via the amplification of
perturbations along the precipitate–matrix
interface and not by the formation of the
matrix phase in the center of the particle.
Thus, the cause of the splitting remains, at
this point, unexplained. However, the
stability of a cuboidal shaped particle in
three dimensions has not been examined,
and the effects of elastic inhomogeneity are
still to be explored fully. Recent work by
Lee (2000) has shown that splitting is pos-
sible for certain differences in elastic con-
stants between particle and matrix.
In addition to inducing particle migra-
tion and selective coarsening, elastic inter-
actions may alter the kinetics of the coars-
ening process. Calculations employing
fixed (spherical or circular) particle mor-
phologies have shown that in a two-particle
system inverse coarsening is possible,
wherein a small particle will grow at the
expense of a large particle (Johnson,1984;www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

432. Miyazaki et al.,1986; Johnson et al., 1990).
However, similar calculations of the evolu-
tion of two precipitates in an elastically an-
isotropic homogeneous system wherein the
morphology is not fixed have shown that
inverse coarsening does not occur (Su and
Voorhees, 1996). The difference between
these two results is due to the strong shape
dependence of the elastic interaction en-
ergy; the particle morphology must be al-
lowed to change in a manner that is consis-
tent with the elastic and diffusion fields in
the system. Inverse coarsening can still oc-
cur, however, when more than two particles
interact elastically (Su and Voorhees, 1996;
Wang et al., 1992). If, however, L is suffi-
ciently large and the system is both elasti-
cally anisotropic and inhomogeneous,
Schmidt et al. (1998) have shown that two
arbitrarily shaped particles can indeed be
stable with respect to coarsening. These
two results imply that it is essential not to
constrain the morphology of the precipi-
tates when computing the evolution of the
microstructure and that elastic inhomoge-
neity can be an important factor in deter-
mining the stability of the system with re-
spect to coarsening. Paris et al. (1995) have
also emphasized the importance of elastic
inhomogeneity in determining the stability
of a system with respect to coarsening.
This implies that determining the evolution
of a system during coarsening with a large
number, sufficient to provide accurate sta-
tistical information, of arbitrarily shaped
particles is a very challenging problem.
Nevertheless, some attempts have been
made. For example, Onuki and Nishimori
(1991) have found through diffuse inter-
face calculations that when the precipitates
are softer than the matrix, stabilization of
the system with respect to coarsening may
be possible. Although the size of the
system examined is small, the exponent of
the temporal power law for the average
particle size is time dependent and, at the
very least, much less than the classical
value of 1/3 predicted by the LSW theory.
In contrast, Ising model simulations (Fratzl
and Penrose, 1996) and diffuse interface
calculations (Nishimori and Onuki, 1990)
in an elastically anisotropic homogeneous
system do not show stabilization with re-
spect to coarsening, again indicating the
importance of elastic inhomogeneity. By
assuming that the systems are statistically
invariant, Leo et al. (1990) have shown that
the exponent for the average size scale of
the precipitates can be altered by the pres-
ence of elastic stress, in this case attaining
380 5 Homogeneous Second-Phase Precipitation
Figure 5-48.Variation of the mean radius R
–
of  g¢-
particles with aging time in two different Ni–Cu–Si
alloys with f
p= 0.18 (Ni–47.4 Cu–5Si) andf
p= 0.5
(Ni–35.1 Cu–9.8 Si), numbers indicate at.%. (After
Miyazaki and Doi, 1989.)
Figure 5-49.Variation of the standard deviation 
sof
the 
g¢-particle size distribution with aging time for
the same alloys as in Fig. 5-48. (After Miyazaki and
Doi, 1989.)www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

433. 5.7 Numerical Approaches to Concomitant Processes 381
a value of 1/2. This result does not contra-
dict Onuki’s, as the onset of stabilization
clearly violates the statistically time invar-
iant assumption of Leo et al.
There have been many investigations of
the kinetics of coarsening in elastically
stressed solids. Experiments performed in
Ni–Cu–Si alloys by Yoshida et al. (1987),
and Miyazaki and Doi (1989) show a clear
departure from the classical t
1/3
predictions
for the average particle radius with the
growth rate of the average particle radius
and the width of the particle size distribution
decreasing in time (Figs. 5-48 and 5-49). A
similar departure from the t
1/3
function has
also been observed in the Ti–Mo system by
Langmayr et al. (1994). In contrast, in
Ni–Al (Ardell, 1990), Ni–Mo–Al (Fähr-
mann et al., 1995) and Ni–Si (Cho and Ar-
dell, 1997) alloys the exponent for the aver-
age particle size appears to be 1/3. The rate
constant, however, appears to be indepen-
dent of the volume fraction of precipitate.
Although it has been speculated that elastic
interactions are responsible for this result,
this has not been confirmed due to the diffi-
culty of performing calculations in systems
that are sufficiently large to yield accurate
statistical information.
5.7 Numerical Approaches
Treating Nucleation, Growth
and Coarsening as Concomitant
Processes
5.7.1 General Remarks on the
Interpretation of Experimental Kinetic
Data of Early Decomposition Stages
In this section we turn our attention 
to decomposition studies of alloys with 
sufficiently high nucleation barriers
(DF*/kTt7) decomposing via nuclea-
tion, growth and coarsening. In contrast to
the considerations of the preceding sec-
tions we now ask how experimental kinetic
data can be interpreted if they refer to very
early decomposition stages which include
nucleation,  growth andcoarsening as con-
comitant rather than consecutive processes
on the time scale. In attempting such inter-
pretations we have to recall that these early
decomposition stages especially are char-
acterized by dramatic changes in supersat-
uration and, thus, in the driving forces for
nucleation and growth processes. Further-
more, during these early stages the size dis-
tribution function evolves and is subjected
to drastic alterations with rather short ag-
ing periods where nucleation, growth and
coarsening must be seen as competing and
overlapping processes. Evidently these
rather complicated phenomena are not
properly accounted for by splitting the
course of decomposition into a nucleation
regime, a growth regime, and a coarsening
regime. Moreover, the kinetic theories de-
veloped for either regime (Secs. 5.5.1, 5.5.2
and 5.6, respectively) are based on ideal-
ized assumptions which are frequently not
expected to be close to reality. This holds
true for the classical nucleation theories of
Volmer and Weber and Becker and Döring
(Sec. 5.5.1). In these theories the supersat-
uration is assumed to be constant. This may
only be fulfilled – if at all – during the ear-
liestnucleation stage. Furthermore, these
theories are based on artificial assumptions
of the cluster size distribution in the vicin-
ity of the critical radius (Fig. 5-24), which
are not consistent with the fact that during
the nucleation process, many growing pre-
cipitates slightly larger than the critical
size are formed. The theory of diffusional
growth by Zener and Ham (Sec. 5.5.2) de-
scribes only the time evolution of precipi-
tates with uniformsize. However, towards
the end of nucleation as well as at the be-www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

434. ginning of the LSW coarsening regime we
certainly have to deal with a polydispersed
precipitate microstructure. Thus, even if
nucleation and LSW coarsening were to 
be well separated on the time scale, in
between these two regimes the Zener–Ham
theory cannot be expected to correctly pre-
dict the measured growth kinetics quantita-
tively. Finally, the coarsening theories of
Lifshitz, Slyozov and Wagner are based on
the linearized version of the Gibbs–Thom-
son equation (Eq. (5-46)) and on the as-
sumption that the supersaturation is close
to zero. These restrictions also hold for the
more recent theories (Sec. 5.6.3) which
take into account finite precipitated vol-
ume fractions and overlapping concentra-
tion profiles between precipitates. 
Thus, any theory for the kinetics of pre-
cipitation that can be employed either for a
more realistic interpretation of experimen-
tal data or for a prediction of the dynamic
evolution of a second-phase microstructure
under elevated temperature service condi-
tions, has to treat nucleation, growth and
coarsening as concomitant processes. This
was accounted for in the cluster dynamics
theories of Binder and coworkers (Secs.
5.5.3, 5.6.4 and Binder and Fratzl (2001),
Chapter 6 of this volume). However, as
was pointed out by these authors (Binder et
al., 1978; Mirold and Binder, 1977), the
theory developed yields a reasonable qual-
itative prediction of the features of experi-
mentally observable quantities but does not
attempt their quantitative interpretation.
A further decomposition theory treating
nucleation, growth, and coarsening as con-
comitant processes was developed by
Langer and Schwartz (LS model; Langer
and Schwartz, 1980). This theory was for-
mulated for describing droplet formation
and growth in near-critical fluids. It was
later modified by Wendt and Haasen
(1983) and further improved by Kamp-
mann and Wagner (MLS model; 1984) in
such a way that it could be applied for the
description of the kinetics of precipitate
formation and growth in metastable alloys
of rather high degrees of supersaturation.
The MLS model is still based on the same
assumptions as the original LS theory. In
particular, the explicit form of the size dis-
tribution is not accounted for and the long-
time coarsening behavior is assumed to
match the LSW results, i.e., is described by
Eqs. (5-63a, 5-65). A priori, it is not pos-
sible to foresee the influence of these as-
sumptions on the precipitation kinetics. 
Therefore, Kampmann and Wagner
(KW; 1984) have devised an algorithm that
accurately describes the entire course of
precipitation within the framework of clas-
sical nucleation and growth theories. This
algorithm is accurate in the sense that, un-
like the LS and the MLS theories, it con-
tains no simplifying assumption; in partic-
ular, in this algorithm, termed the Numeri-
cal model (N model), the time evolution of
the size distribution is computed without
any approximations. From a comparison of
the N model with experimental data it is
possible to determine some crucial precipi-
tation parameters of the particular alloy
system as well as to scrutinize the existing
nucleation and growth theories with re-
spect to their applicability to decomposi-
tion reactions in real materials. Further-
more, the N model allows an evaluation to
be made of how realistic the various ap-
proximations are which occur in both the
LWS theory and the MLS model.
In this section we briefly introduce the
MLS and the N models. An attempt is
made to demonstrate their ability to de-
scribe the entire course of the decomposi-
tion reaction; we will also show how some
essential parameters of the decomposing
alloy, such as the diffusion constant and the
specific interfacial energy, can be evalu-
382 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

435. 5.7 Numerical Approaches to Concomitant Processes 383
ated by fitting the theoretically predicted
curves R
–
(t) and N
v(t) to the corresponding
experimental data. Furthermore, on the
grounds of the N model, it is possible to ex-
amine whether certain time intervals exist
during the course of a precipitation reac-
tion in which the kinetic evolution is pre-
dicted by either the growth or the coarsen-
ing theory with sufficient accuracy.
5.7.2 The Langer and Schwartz Theory
(LS Model) and its Modification
by Kampmann and Wagner (MLS
Model)
In the LS theory it is assumed that the
system contains N
LSdroplets per unit vol-
ume of uniform size R
–
LS. In order to ac-
count for coarsening, the continuous distri-
bution function f(R,t) and the number of
particles of critical size, i.e., f(R*,t) dR*,
must be known. However, in the LS theory
this is not the case. LS, therefore, intro-
duced an apparent density f
a(R*,t) (Fig. 5-
50) which is given as:
(5-66)
f
a(R*,t) is thus proportional to N
LSand in-
versely proportional to the width off(R,t).
The constant b= 0.317 is chosen in such a
way that for tÆ•the coarsening rate
dR
–
3
LS
/dtis identical to K
R
LSW(Eq. (5-65)).
Unlike the LSW theory, where R
–
=R*, in
the LS theory only particles with R>R*are
counted as belonging tof(R,t), i.e.
(5-67)
Thus, R
–
LS>R* at all stages of decomposi-
tion, keepingf
a(R*,t) in Eq. (5-66) finite.
Due to nucleation at a rate Jand dissolu-
tion, N
LSchanges with time according to:
(5-68)
d
d
d
d
LS
aN
t
JfRt
R
t
=– ( *, )
*
R
N
fRtRR
R
LS
LS
d=
∞
∫
1
(,)
*
fRt N
b
RR
LS
aLS
(*,)
–*
=
The growth of particles of mean size R
–
LSis
given as:
(5-69)
The term v(R
–
LS) is given by Eq. (5-43) and
accounts for the growth rate of the particles in the supersaturated matrix. The second term accounts for the change in the true distribution function caused by the dissolu- tion off
a(R*,t) dR* particles with radii
between R* and R* +dR*. The third term
describes the change inf(R,t) caused by
the nucleation of particles which must be slightly larger than those of critical size,
d
d
d
d
LS
LS
LS
a
LS
LS
LSR
t
R
RR
fRt
N
R
t
N
JR t R R R
=
+
++
v()
(–*)
(*,) *
[*()](* *– )
1
d
Figure 5-50.Relationship between the ‘true’ contin-
uous size distribution function f(R,t) yielding the
mean radius R
–
and the related parameters of the LS
model. LS assumed a monodispersive distribution,
f
LS, of particles with radius R
–
LS. f
a(R*,t)dR* is the
apparent number density of particles with radii
between R* and R*+dR*. In the LS theory, only par-
ticles in the hatched region belong to the precipitated
phase.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

436. i.e., R=R*+ dR* and dR*OR*. Together
with the continuity equation
3
,
(5-70)
and Eq. (5-66), Eqs. (5-68) and (5-69) are
the rate equations which describe the entire
course of precipitation in the LS model.
After proper scaling, these equations were
numerically integrated in conjunction with
steady state nucleation theory (Eq. (5-27)).
The LS modelis based on the assumption
that the equilibrium solubility of small
clusters can be determined from the linear-
ized version of the Gibbs–Thomson equa-
tion, i.e. Eq. (5-46). This linearization,
however, generally does not hold for small
clusters in metallic alloys (cf. Sec. 5.6.2).
This becomes immediately evident for the
Cu–1.9 at.% Ti system isothermally aged
at 350°C. At t= 0, R* ist 0.48 nm or 0.13
nm, depending on whether R* is computed
(
s
ab= 0.067 J/m
2
, see Sec. 5.7.4.4) from
the non-linearized or from the linearized
version of Eq. (5-46). This example clearly
demonstrates that Eq. (5-46) must be used
in its non-linearized version, particulary
for systems with large values of 
s
aband/or
large supersaturations.
The MLS model is based on the non-line-
arized Gibbs–Thomson equation (Eq. (5-
45)). Thus, the growth rate in the MLS
model is:
(5-71)
In order to write the rate equations (5-68)
to (5-70) in a properly scaled version, we
introduce the following parameters:
d
d
LS
LS
p
e
LS
e m
gLSR
t
R
cc
D
R
cc
V
RT R
==
×
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
v
ab
()
–
– exp
1
21
a
a
s
(–) (–)cc RN cc
pL SLS
4
3
3
0p
=
(5-72)
Apart from R
N, all parameters are dimen-
sionless; unlike in both the original LS
model and the study of Wendt and Haasen
(1983), in the MLS model R
Nrather than
the correlation length is used as the scaling
length. By straigthforward scaling of Eqs.
(5-27), (5-61) and (5-71) we obtain the
scaled version of the equations of motion
(5-68) and (5-69). From these, the number
density nof particles is eliminated by vir-
tue of the scaled continuity equation
(5-73)
We finally obtain
(5-74)
(5-75)
Eqs. (5-74) and (5-75) are the basic equa-
tions of the MLS model and are numeri-
cally integrated with 
s
0= 0.1 J/m
2
; values
of 
dr*/r*(x
0) used by Kampmann and
Wagner (1984) ranged from 0.05 to 0.2.
In the LS theory only the steady-state
nucleation rate has been used; in contrast,
d
d
d
d
p
p
r
txx
x
t
xr
xr
r
x
xx x
dr r
+
=
++
⎛
⎝
⎜
⎞
⎠
⎟
bk
k
J
k
s
s
s
(ln )
–
[ – exp ( / )]
–
–
˜
ln
*–
2
3
0
1
1
1
1
d
d
d
d
p
r
t
r
xxrx xx
x
t
rxx
xx
++
⎛
⎝
⎜
⎞
⎠
⎟
=
3
11
3
0
4
0–ln–ln
–
˜
–
–
bk
k
J
s
s
n=
1
1
3
0
r
xx
x
–
–
p
R
V
RT
kcc
cc c c R R
RR RR
nN R
D
R
tJJ
n
N
t
N
m
g
e
e
pp
e
LS N
NN
LS N
N
2
d
d
d
d
===
===
==
===
2
4
3
0
3
s
ssx
x xr
r dr d
p
t
t
s
0
ab 00 a
aa
;/;/;
/; /; / ;
* */; * */;
;;
˜
384 5 Homogeneous Second-Phase Precipitation
3
Note that particles with R=R* are not contained in
Eq. (5-70); we therefore call these particles ‘apparent’.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

437. 5.7 Numerical Approaches to Concomitant Processes 385
in Eqs. (5-74) and (5-75) of the MLS
model, KW employed the time-dependent
nucleation rate J* (Eq. (5-29)), which in-
volves the incubation time. In the scaled
version the latter (
t
w) was evaluated from:
(5-76)
When deriving this equation, it was as-
sumed that the minimum time (
t
w, min) for a
particle to reach the critical size 
r* is given
by Eq. (5-47). However, for subcritical nu-
clei c
Ris considerably larger than c
e
a
, and
the probability of their redissolution is
rather large; hence, Eq. (5-47) overesti-
mates the growth rate significantly. This
fact is counterbalanced by the introduction
of the parameter c
w. For each particular al-
loy system, c
wis determined in such a way
that after a period 
t
wthe first particles be-
come ‘observable’ with a reasonable number
density. For all alloy systems investigated so
far, c
wranges from 1.4 to 3.5. Hence, be-
cause of the stochastic nature of cluster
growth and dissolution, on average it takes
about two to twelve times longer for a clus-
ter to attain a size R>R* beyond which its
further growth may be evaluated in a deter-
ministic manner on the basis of Eq. (5-47).
5.7.3 The Numerical Model (N Model)
of Kampmann and Wagner (KW)
Unlike the MLS model, in the N model
the time evolution of f(R,t) – orf(
r,t) if
we stick to the same nomenclature – is
computed. For this purposef(
r,t) is sub-
divided into intervals [
r
j+1, r
j] with
|
r
j–r
j+1|/r
j∫1 and n
jparticles in the j-th
interval. In contrast to the MLS model, dis-
solving particles with 
r<r* are assumed
to belong to the precipitating phase, i.e.
nn
n
n
j
j
j
jj
j
j==
==
∑∑and
11
00
1
rr
tx
x
x
rt
w cc()
–
–
*
, min=≅
1
2
1
1
22 2p
ww w
(5-77)
The continuity equation in the N model
then reads:
(5-78)
Thus the continuous time evolution of
f(
r,t) is split into a sequence of individual
decomposition steps; these step are chosen
in such a way that within each correspond-
ing time interval D
t
ithe changes of all ra-
dii 
r
j,(t
i) and of the supersaturation x(t
i)
remain sufficiently small. Then, both the
nucleation and the growth rates can be con-
sidered as being constant during D
t
iand
the change off(
r,t) and of xcan be reli-
ably computed. The N model of KW con-
tains some algorithms which ensure a
rather high accuracy of the numerical cal-
culation; it amounts to ≈0.5% in the case
of the growth rate of 
r.
5.7.4 Decomposition of a Homogeneous
Solid Solution
5.7.4.1 General Course of Decomposition
KW discussed the decomposition reac-
tion of an ideally quenched alloy with c
a
e=
0.22 at.%, c
p= 0.20 at.%, c
0= 1.9 at.%
and 
s
ab= 0.067 J/m
2
. Just on the basis of
these input data, the decomposition reac-
tion can be calculated within the frame-
work of the MLS and N models. Fig. 5-51
shows the predictions from both the MLS
(full lines) and the N models (discrete sym-
bols) for the time evolution of the radii 
r
–
(
t) and r*(t), the number density n( t),
the supersaturation 
x(t), and the nuclea-
tion rate J
˜
(
t), The chosen values of x
0=
8.7, c
a
e,sand Dcorrespond to those for
Cu–1.9 at.% Ti aged at 350°C (see Secs.
5.7.4.3 and 5.7.4.4). Surprisingly, both
models yield qualitatively similar results.
(–) –xrxx
p1
1
3
0
0
j
j
jj
n
=
∑ =
jjj=+
+with
1
1
2
rrr()www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

438. During the early nucleation period (tÙ1)
both J
˜
and nincrease, whereas 
xand,
hence, 
r*remain roughly constant. At this
stage, the N model gives a rather narrow
size distribution functionf(
r,t) (Fig.
5-52). Since c
Ris still close to c
–
≈c
0, the
growth rate of nucleated precipitates is also
close to zero, i.e., 
r
–
remains about con-
stant. After (
tÛ1, those precipitates nucle-
ated first become considerably larger than
r*andf( r,t) becomes much broader. This
is the beginning of the growth period
386 5 Homogeneous Second-Phase Precipitation
Figure 5-51.Evolution of various scaled precipitation parameters with scaled aging time taccording to both the
N model and the MLS model. The chosen values of 
x
0, s
abcorrespond to those for Cu–1.9 at.% Ti aged at 350°C.
Figure 5-52.Evolution of the scaled size distribution function in Cu-1.9 at.% Ti with aging times as computed
with the N model. For comparison, the distribution functionsf
LSW(r,t) of the LSW theory with the known 
values of N
v(t) and R
–
(t) are also shown. D=2.5×10
–15
cm
2
/sec.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

439. 5.7 Numerical Approaches to Concomitant Processes 387
which is characterized by (i) the highest
growth rate ever observed during the
course of precipitation; (ii) a ratio 
r
–
/
r*
which becomes significantly larger than 1;
(iii) the maximum number density (n
max,
N
v,max) of particles which remains about
constant; (iv) a stronger decrease of 
x; and,
consequently, (v) a decrease of J
˜
from its
maximum value; in this particular case, J
˜
never reaches its steady-state value. At the
end of the growth regime (
t≈15) the
supersaturation has dropped significantly
and the growth rate becomes small. This
effect causes 
r*to converge towards r
–
and
makes dn/d
t< 0 (Eq. (5-68)). During the
subsequent transition period (
t≈15), the
growth rate of 
r
–
is primarily controlled by
the dissolution of particles with 
r<r* and
only to a lesser extent by the uptake of so-
lute atoms from the matrix, the supersatu-
ration of which is still about 20%. During
the transition period, the true distribution
functionf(
r,t) continuously approaches
the one predicted by LSW (Fig. 5-52). At
this stage, however,f
LSWis still a rather
poor approximation forf(
r,t). This simply
reflects the influence of the linearization of
the Gibbs–Thomson equation (Eq. (5-46))
on which the LSW theory is based. For
tÆ•, dr
–
3
/dtapproaches asymptotically 
a constant value, i.e., the reaction is within
the asymptotic limit of coarsening where
the supersaturation is almost zero. At this
stagef(
r,t) is well approximated byf
LSW
with only minor deviations for small parti-
cle radii.
5.7.4.2 Comparison Between the MLS
Model and the N Model
The precipitation reaction starts with an
identical nucleation rate and identical par-
ticle sizes in both the MLS and the N mod-
els. Therefore, both models are expected to
yield identical results which, in fact, is ob-
served in Fig. 5-51. During the later
growth period, dissolution of particles with
r<r*commences. At this stage, the MLS
model only counts particles with 
r>r*as
belonging to the second phase (Eq. (5-67)),
and, furthermore, assumes f
a(r=r*)to be
proportional to 1/(
r
–
–
r*) (Eq. (5-66)). At
this stage, this is a rather poor approxima-
tion because the N model yields a steep
slope forf(
rÆr*,t) with a rather small
density n(
r=r*).These facts make the
MLS model predict considerably larger rate
constants for the decrease of the particle
number density and, hence, for the growth
of 
r
–
at the end of the growth period (
t≈15). 
Depending on the particular choice of b
(Eq. (5-66)), the coarsening rate of the
MLS model approaches asymptotically the
value K
R
LSW(Eq. (5-65)) from the LSW the-
ory. In this asymptotic limit the mean radii
from both models and, hence, their coars-
ening rates become identical. At this stage
the N model yields 
r
–
Ó
r*as predicted by
LSW, whereas the MLS model yields 
r
–
–
r*= const., i.e., R
–
LS>R*, as required by
Eq. (5-67).
We can conclude that the MLS model,
which requires much less computing time
than the N model, provides a good survey
of the general course of precipitation.
However, due to the simplifying assump-
tions made, it does not predict the precipi-
tation kinetics with the same accuracy as
the N model. This is particularly evident
for those precipitation stages where the
shape off(
r,t) is extreme, as, for in-
stance, during the later growth stages in the
example discussed above.
5.7.4.3 The Appearance and
Experimental Identification
of the Growth and Coarsening Stages
In the following, the results from the
MLS and the N models are compared withwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

440. 388 5 Homogeneous Second-Phase Precipitation
Figure 5-53.Variation of R
–
,
R*, R
–
LSand of the relative
supersaturation with aging
time for Cu–1.9 at.% Ti as
computed with the N model
and the MLS model for the
given set of thermodynamics
data; also shown are the ex-
perimental data for R
–
(t) from
von Alvensleben and Wagner
(1984). 
Figure 5-54.Variation of R
2
and R*
2
with time during the growth regime for Cu–1.9 at.% Ti. During the period
marked by the two arrows the kinetics follow the power-law R
–
t
1/2
.
b) Variation of the coarsening rate dR
–
3
/dtwith aging time; K
R
LSWis the value predicted by the LSW theory,
Eq. (5-65).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

441. 5.7 Numerical Approaches to Concomitant Processes 389
experimental data obtained from a FIM
atom probe study of early-stage precipita-
tion in Cu–1.9 at.% Ti at 350°C (von Al-
vensleben and Wagner, 1984).
Fig. 5-53 shows the time evolution of R*
and or R
–
in physical units. Again these ex-
perimental data points  are well described
by the N model. Fig. 5-53 also reveals that
the experimental data cover neither the
early nucleation period nor the growth re-
gime, but rather start (t = 2.5 min) at the
end of the latter region. From Fig. 5-54a,
where R
–
2
is plotted as a function of t, it is
recognized that during the period t≈
0.7 min to 1.2 min, R
–
2
varies linearly with t
as suggested by Eq. (5-40). Since this time
period extends only over 0.5 min, KW con-
cluded that for Cu–1.9 at.% Ti the time
window is too short for revealing the 
R
–
∂t
1/2
kinetics experimentally; in fact, the
same holds true for many other alloy
systems analyzed by KW using the N
model. Furthermore, from Fig. 5-54a the
slope of the straight line has been evaluated
to be 0.88 nm
2
/min, whereas the corre-
sponding growth rate from Eq. (5-47) is
computed (c
a
e= 0.22 at.%, c
p= 20 at.%, 
D= 2.5¥10
–15
cm
2
/sec) to be 2.5 nm
2
/
min. This result clearly demonstrates that
no growth regime exists which is ade-
quately accounted for by Eq. (5-47). In
other words, the idealizations made in the
derivation of Eq. (5-47) do not approxi-
mate the true situation in Cu–1.9 at.% Ti.
However, if c ¯(t) and c
Rin Eq. (5-43) are
replaced by their mean values respectively,
during the period for which R
–
∂t
1/2
holds,
then Eqs. (5-40 and 5-42) with 
l
i=∂≈k*
yield a value for the growth rate (0.104 nm
2
/
min) which is only 16% higher than the
true value.
In Fig. 5-54b the rate constant dR
–
3
/dtfor
coarsening is plotted versus t. It is evident
that for Cu–1.9 at.% Ti the rate constant
K
R
LSW= 1.2¥10
–24
cm
3
/sec from the LSW
theory (Eq. (5-65)) is only reached for ag-
ing times beyond ≈10
4
min! At this stage  R
–
has already grown to ≈6.4 nm. From this
result, which reflects the influence of the
linearization of Eq. (5-46), KW inferred
that the LSW theory predicts the correct
coarsening rate once 
(5-79)
If this relation holds,f(R,t) is almost iden-
tical tof
LSW(Fig. 5-52) and cˉ/c
a
e≈1.
According to the results from the N
model, during the early coarsening stages
the precipitation kinetics deviate signifi-
cantly from those predicted by the LSW
theory (e.g., Fig. 5-54b). Thus the widely
used LSW analyses resulting in a determi-
nation of Dand 
s
abon the basis of Eq. 
(5-63a) and Eq. (5-63c) should not be ap-
plied to early coarsening stages during
which the relation does not hold.
5.7.4.4 Extraction of the Interfacial
Energy and the Diffusion Constant
from Experimental Data
KW determined 
s
aband Dseparately by
fitting R
–
(t) and N
v(t) as obtained from the
N model to the corresponding experimental
data. Fig. 5-55 shows the variation of N
v
and J* with aging time  as computed with
the N and the MLS models, together with
the corresponding experimental data. The
peak number density N
v, maxof particles in
a precipitation reaction is essentially gov-
erned by the value of DF* via the nuclea-
tion rate equation (5-27). Since DF*∂
s
3
ab
(Eq. (5-20)), N
v, maxdepends sensitively on
the value of the interfacial energy 
s
ab. The
very strong dependence of N
v, maxon s
abis
clearly revealed by Fig. 5-55, showing a
good fit of N
v(t) for s
ab= 0.067 J/m
2
and
only poor agreement for 
s
ab= 0.071 J/m
2
.
On the other hand, a variation in Dmani-
R
V
RT
≈13
m
gs
abwww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

442. fests itself in a parallel shift of the entire
curve on the time scale. Thus, a fit of R
–
(t)
and of N
v(t) as obtained from the N model
to the experimental curves allows both 
s
ab
and D to be determined quite accurately. 
From a variety of different two-phase al-
loys, the available kinetic data have been
interpreted in terms of the N model. Table
5-3 presents the interfacial energies 
s
abto-
gether with the width of the (coherent) mis-
390 5 Homogeneous Second-Phase Precipitation
Figure 5-55.Variation of N
vand J* with aging time for Cu–1.9 at.% Ti as computed with the N model and the
MLS model for 
s
ab= 0.067 J/m
2
; for this value of s
abthe computed N
v(t) curve agrees well with the experi-
mental data; poor agreement is obtained for 
s
ab= 0.071 J/m
2
.
Table 5-3.Correlation between the width of the miscibility gap and the coherent interfacial energies 
s
abfor
various two-phase alloys as determined from a fit of the N model to experimental kinetic data.
Alloy Aging Composition/type Width of coherent Coherent interfacial
temperature of precipitates miscibility gap energy
at.% °C at.% s
abJ/m
2
Ni–14 Al
1
550, 500 g¢-Ni
3Al ≈15 ≈0.016
Ni–26 Cu–9 Al
2
550, 500 g¢-(Cu, Ni)
3Al ≈20 ≈0.052
540, 500 Not determined ≈20 ≈0.050
580, 500 Not determined ≈20 ≈0.052
Cu–1.9 Ti
3
350, 500 b¢-Cu
4Ti ≈20 ≈0.067
Cu–2.7 Ti
4
350, 500 b¢-Cu
4Ti ≈20 ≈0.067
Cu–1.5 Co
5
500, 500 >95 at.% Co ≈95 ≈0.171
Fe–1.4 Cu
6
400, 500 >98 at.% Cu ≈100 ≈0.250
Fe–0.64 Cu
6
400, 500 >98 at.% Cu ≈100 ≈0.250
1 Wendt and Haasen (1983) (AFIM)
2 Liu and Wagner (1984) (AFIM)
3 Alvensleben and Wagner (1984) (AFIM, CTEM)
4 Eckerlebe et al. (1986) (SANS)
5 Gust (1986), unpublished (magnetic measurements)
6 Kampmann and Wagner (1986) (SANS)www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

443. 5.7 Numerical Approaches to Concomitant Processes 391
cibility gap for each given alloy. It is evi-
dent that there is a pronounced correlation
between 
s
aband the compositional width,
i.e., the broader the gap, the larger is 
s
ab.
This is consistent with various theoretical
predictions on the interfacial energy (see
Lee and Aaronson, 1980, for a comprehen-
sive discussion of this aspect).
5.7.5 Decomposition Kinetics in Alloys
Pre-Decomposed During Quenching
The versatility of the N model is further
illustrated by its ability to predict the pre-
cipitation kinetics in alloys which have ex-
perienced some phase separation during
quenching. This is exemplified for Cu–2.9
at.% Ti, the decomposition reaction of
which was studied by Kampmann et al.
(1987) by means of SANS. They found that
the cooling rate of their specimen was not
sufficient to suppress the formation of
Cu
4Ti precipitates during the quench; in
fact, as is shown in Fig. 5-56, the solute
concentration decreased from c
0= 2.9 at.%
to 2.2 at.% Ti during the quench. With fur-
ther aging, the supersaturation decreased
continuously through the formation of ad-
ditional clusters and the growth of existing
ones. After aging for ≈ 100 min at 350°C
the metastable b¢-solvus line is nearly
reached at c¢
a
e≈0.22 at.% Ti (cf. Sec. 5.2.1).
The experimentally determined kinetic
behavior of the precipitate number density,
of their mean radius, and of the supersatu-
ration are displayed in Fig. 5-57a–c and
compared with the predictions of the N
model. For the computations, the result
from the SANS evaluation was taken into
account, which yielded the homogenized
sample to already contain ≈2¥10
25
clus-
ters/m
3
with R
–
≈0.7 nm. These could im-
mediately grow by further depleting the
matrix from solute atoms. Moreover, at 
t= 0 the supersaturation was still large
enough for nucleating new clusters with
smaller radii at a nucleation rate J* (Fig.
5-57a). Thus, during the first minutes of
aging, the cluster number density in-
creased. At this stage, the alloy contained a
sort of bimodal cluster distribution: the
larger ones formed at a smaller supersatu-
ration during quenching, and the smaller
ones resulted from nucleation at 350°C.
Due to both nucleation of new clusters and
growth of pre-existing ones, the supersatu-
ration and, hence, the nucleation rate de-
creased rapidly; after aging for ≈3 min, nu-
Figure 5-56.Decrease in the solute con-
centration in the matrix with aging time, as
determined from Laue scattering (∫∙), from
the integrated intensity (¥), and the Guinier
approximation (≤) of SANS curves.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

444. cleation is virtually terminated. After about
10 min the critical radius R*, which is cor-
related with the momentary supersatura-
tion, reaches the mean value R
–
of the glo-
bal size distribution. At this instant, R* has
grown beyond the mean radius of the
smaller, freshly nucleated clusters; these
now redissolve leading to a further de-
crease in N
v. Now  the size distribution is
again governed by the larger precipitates
formed during quenching. After about
500 min, N
vdecreases with tas expected
from the LSW theory. 
With regard to the accuracy of both the
SANS experiment and, in particular, the
SANS data evaluation, the agreement
between the experimental kinetic data and
those from the N model is rather good for
an interfacial energy 
s
ab= 0.067 J/m
2
.
This value is identical to that determined
for the less concentrated Cu–1.9 at.% Ti al-
loy (Sec. 5.7.4.4). In the early stages (t≤
10 min) the experimental R
–
is considerably
larger than the theoretical one. This simply
reflects the fact that the scattering power of
a particle is weighted with R
6
; hence, for
t≤10 min, essentially the radius of only
the larger particles within the bimodal dis-
tribution was determined. The diffusion 
coefficient (D =3¥10
–16
cm
2
/sec) was
found to be about one order of magnitude
smaller than in the Cu-1.9 at.% Ti alloy
(Sec. 5.7.4.4). It was not possible to decide
whether this difference in the Dvalues re-
flects the concentration dependence of D,
or whether it was caused by differences in
the homogenization temperatures of the
two alloys (Cu–2.9 at.% Ti: T
H= 780°C;
Cu–1.9 at.% Ti: T
H= 910°C).
5.7.6 Influence of the Loss of Particle
Coherency on the Precipitation Kinetics
In many two-phase systems the particles
lose coherency once they have grown be-
392 5 Homogeneous Second-Phase Precipitation
Figure 5-57.Time evolution of (a) the cluster num-
ber density N
vand the nucleation rate J*, (b) of their
mean radius R
–
and of the critical radius R*, and (c) of
the supersaturations. Experimental SANS results:
discrete symbols; computational results: full lines.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

445. 5.7 Numerical Approaches to Concomitant Processes 393
yond a certain size R
T. The associated in-
crease in the interfacial energy and de-
crease in the solubility limit (cf. Fig. 5-1)
leads to an enhanced driving force and,
hence, to accelerated kinetics for further
coarsening of the incoherent microstruc-
ture with respect to the coherent one. This
effect can also be accounted for by the N
model. This is shown in Fig. 5-58 for
Fe–1.38 at.% Cu aged at 500°C. As in-
ferred from CTEM, the Cu-rich particles
transform at R
T≈2.8 nm from the meta-
stable b.c.c. structure into the f.c.c. equilib-
rium structure. The associated loss of cohe-
rency occurs at particle number densities
well beyond the maximum number density 
N
v, max(Fig. 5-58). Thus, following the
procedure outlined in Sec. 5.7.4.4, the co-
herent interfacial energy could be deter-
mined by fitting the N model to the experi-
mental (SANS) kinetic data of the still co-
herent system. The value 
s
ab≈0.27 J/m
2
obtained is considerably smaller than the
corresponding value 
s
inc
ab
≈0.50 J/m
2
which was derived from a thermodynami-
cal analysis of the Fe–Cu system (Kamp-
mann and Wagner, 1986). As is shown in
Fig. 5-58, the loss of coherency in fact
leads to a momentary acceleration of the
growth kinetics. It is, however, not suffi-
cient to bridge the discrepancy between the
experimental kinetic data and the theoreti-
cally predicted ones displayed in Fig. 5-58.
As the predicted kinetics are much more
sluggish than the experimentally deter-
mined ones, we might speculate that ne-
glecting particle interaction accounts for
the observed discrepancy. Inspection of
Fig. 5-43, however, reveals that a consider-
ation of finite volume effects in Fe–1.38
at.% Cu withf
p≈1% would increase the
coarsening rate only by less than a factor
1.3 whereas a factor of ≈10 is required to
match the results from the N model and the
SANS experiments at the later stages of
Figure 5-58.Kinetic evolution of the precipitated
number density (top) and of the mean radius (bottom)
as predicted by the N model for Fe–1.38 at.% Cu.
The solid points refer to experimental data derived
from nuclear and magnetic SANS experiments. The
dashed lines show the kinetic evolution without con-
sideration of the b.c.c.
Æf.c.c. transformation of the
copper-rich particles.www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

446. precipitation. At present, it is unclear
whether the various theories dealing with
finite volume effects (Sec. 5.6.3) are still
insufficient or whether some heterogenous
precipitation at lattice defects accounts for
the observed discrepancy.
At first glance the experimental kinetic
data for R
–
(t) and N
v(t) in Fig. 5-58 might
be seen as being amenable to an LSW anal-
ysis in terms of Eq. (5-63). Analyses of the
SANS data for t≤10
3
min, however,
showed the width of the particle size distri-
bution to be much broader (standard devia-
tion: 
s≈0.31) than expected from the 
LSW theory or its modification (
s≈0.23).
Furthermore, the measured supersaturation
was still far from being close to zero.
Hence, the conditions for an LSW analysis
of the experimental data are not at all ful-
filled.
5.7.7 Combined Cluster-Dynamic
and Deterministic Description
of Decomposition Kinetics
There remain two main shortcomings of
the N model outlined in Sec. 5.7.3. Firstly,
the stochastic nature of the nucleation pro-
cess is solely accounted for by using the
time- and concentration-dependent nuclea-
tion rate (i.e., Eq. (5-29)), whereas the
growth (and shrinkage!) is computed in a
deterministic manner on the basis of Eq. 
(5-43). Thus, the stochastic nature of early-
stage growth which, in particular, becomes
effective in systems with small nucleation
energies, is not adequately accounted for in
the N model.
Secondly, as in all previous theories, the
matrix concentration c
Rat the cluster–ma-
trix interface was also calculated on the ba-
sis of the Gibbs–Thomson equation, Eq.
(5-46). This relation, which describes an
equilibrium between the cluster size Rand
the matrix concentration c¯, is generally not
394 5 Homogeneous Second-Phase Precipitation
adequate for describing the growth (R>R*)
or dissolution (R<R*) of clusters.
To overcome these shortcomings, Kamp-
mann et al. (Kampmann et al., 1992; Staron
and Kampmann, 2000 a, b) extended the N
model by a cluster-dynamics (C–D) simu-
lation of the kinetics during the nucleation
and growth stages (cf. Sec. 5.5.3). To save
computation time, in the later growth and
coarsening stages where the stochastic pro-
cess is no longer relevant, the cluster-dy-
namics approach is linked to the determin-
istic description of the original N model.
As in the latter, the input parameters enter-
ing the extended C–D model are c
0,c
a
e, s
and Dwith variations in Dleading only to
shifts of the computed curves (e.g., R
–
(t) or
J(t)) parallel to the time scale. Therefore,
this C–D simulation can be used in as ver-
satile a manner as the original N model 
for the interpretation of experimental data 
(Staron and Kampmann, 2000b).
The cluster-dynamics algorithm was de-
vised such that its corresponding Helm-
holtz energy functional approximates
closely that of the regular solution model.
As proved by CALPHAD studies, this is 
a rather good approximation for Cu–Co.
Fig. 5-59 shows the computed time evolu-
tion of the cluster size distribution during
the nucleation period and the nucleation
rate with input parameters corresponding
to a Cu–0.8 at.% Co model alloy aged at
500°C. At t= 5 sec all clusters are still be-
low the critical size
4
and, thus, Fig. 5-59a
provides insight into the incubation process.
The nucleation rateJ
S
B–D
calculated ac-
cording to Eq. (5-27) with c¯∫c
0exceeds
the maximum computed one (Fig. 5-59b)
by more than a factor of 1000. This dis-
crepancy cannot be attributed to incubation
4
In the C–D approach the critical radius is defined
as that size for which the probabilities of cluster
growth and shrinkage are identical.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

447. 5.8 Self-Similarity, Dynamical Scaling and Power-Law Approximations 395
effects; it can only be resolved if – instead
of Eq. (5-27) – a more adequate equation is
used for calculating the Becker–Döring nu-
cleation rate:
(5-80)
The modified exponent takes into account
that no Helmholtz energy D F* (1) is
needed for forming monomers (which are
already dissolved in the matrix during the
solution treatment).
Accordingly the number of lattice points
(N
0) in Eq. (5-28) must be replaced by the
number of monomers n
1(t), and for a calcu-
lation of DF* the concentration of
monomers c
1(according to Fig. 5-60, c
1=
0.7 at.% rather than the nominal concentra-
tion of c
0= 0.8 at.%) must be used. The
factor 
1
∕2in Eq. (5-80) takes into account
that monomers serve both as initial clusters
(containing i= 1 solute atom) capable of
growing, and also as reaction species being
able to initiate the growth of “monomer
clusters”. As shown in Fig. 5-59b, with
these physically justified modifications,
excellent agreement is obtained between
the maximum nucleation rate (J
max) as ob-
tained from the cluster-dynamics model,
andJ
ˆ
S
B–D
(Eq. (5-80)).
5.8 Self-Similarity,
Dynamical Scaling
and Power-Law Approximations
5.8.1 Dynamical Scaling
According to the LSW theory and its ex-
tensions to finite volume fractions (Sec.
5.6), the distribution of relative particle
sizes R/R
–
evolves during extended aging
(tÆ•) towards an asymptotic, time-invar-
iant form, the particular shape of which de-
ˆ
*()
exp {– [ *( ) – *( )]/ }
JZnt
Fc F kT
B–D
S=
×
1
2
1
1
1b
DD
Figure 5-59.a) Evolution of the cluster size distri-
bution during the nucleation stage for a Cu–0.8 at.%
Co model alloy aged at T= 500°C, with pair ex-
change parameter 
W= 6.24, c
a
e= 0.2 at.%, D
CuCo=
1¥10
–19
m
2
/s.
b) Time evolution of the nucleation rate J(t) (solid
line) according to the cluster-dynamics calculation.
J
S
B–D
is based on Eq. (5-27), Jˆ
S
B–D
is the modified
Becker–Döring nucleation rate (Eq. (5-80)). (After
Kampmann et al., 1992.)www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

448. pends on the precipitated volume fraction
(Fig. 5-44). The final time invariance of 
f(R/R
–
) reflects the fact that once the pre-
cipitated volume fraction has reached its
equilibrium value, consecutive configura-
tions of the precipitate microstructure are
geometrically similar in a statistical sense,
i.e., all consecutive configurations are sta-
tistically  uniform on a scale that is consid-
erably larger than some characteristic
length such as the mean particle size R
–
or
the mean center-to-center distance L
–
=
N
v
–1/3. The self-similarity of the microstruc-
tural evolution has found its expression in
the dynamical scalingof the structure func-
tion S(
k,t) (Binder and Stauffer, 1974;
Binder et al., 1978). Furukawa (1981) pro-
posed S(
k,t) to satisfy (after some tran-
sient time t
0) a scaling law of the form:
S(
k,t) =l
3
(t)F
˜
[kl(t)];t≈t
0(5-81)
where F
˜
[
kl(t)]∫F
˜
(x) is the time-inde-
pendent scaling function. As the scaling
parameter, l(t) denotes some characteristic
length and contains exclusively the time
dependence of S(
k,t).
Strong theoretical support for the valid-
ity of the scaling hypothesis, Eq. (5-81),
during the later stages of decomposition
was first provided by Monte Carlo simula-
tions of the time evolution of binary model
alloys (Sec. 5.5.6). From these studies it
was concluded that there is a small though
systematic dependence ofF
˜
(x) on the in-
itial supersaturation, at least for small x
(Lebowitz et al., 1982); for large values of
x, the scaling function appears to be uni-
versal in that it becomes independent of
temperature and precipitated volume frac-
tion and even of the investigated material 
(Fratzl et al., 1983). By analogy to the
Porod law of small-angle scattering, in this
regimeF
˜
(x) decays in proportion to x
–4
.
In experiments designed to test the valid-
ity of the scaling behavior, l(t) is com-
monly related to either the radius of gyra-
tion R
G, the mean particle radius, or to ei-
ther 
k
m
–1(t) or k
1
–1(t) if k
mand k
1denote
the maximum and the first moment of S(
k,
t), respectively. The scaling functionF
˜
(x)
is then simply obtained, for instance, by
plotting 
k
m
3S(k,t) versus k/k
m. If scaling
holds, F
˜
(
k/k
m) is time independent
5
. After
396 5 Homogeneous Second-Phase Precipitation
Figure 5-60.Time evolution of the
number density (n
v), the mean (dy-
namical) critical radius (R*), the
mean radius of supercritical clus-
ters (R
–
); the conventionally defined
supersaturation 
x–1 =c¯/c
e
a
–1 is
compared with the supersaturation
x
mono–1 = c
1/c
e
a
–1 of monomers.
For t≤100 min the computation
was performed cluster-dynamically
and afterwards continued determin-
istically on the basis of the N
model of Sec. 5.7.3. (After Kamp-
mann et al., 1992.)
5
In order to test whether experimental data satisfy
the scaling law, Fratzl et al. (1983) have proposed a
direct method by which the evaluation of R
G, R
–
, k
m
or k
1can be avoided and by means of which F˜(x)
can be determined graphically.www.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

449. 5.8 Self-Similarity, Dynamical Scaling and Power-Law Approximations 397
some initial transient time this, in fact, was
observed in the glass systems B
2O
3–PbO–
Al
2O
3(Craievich et al., 1981, 1986) and in
several binary alloys such as Mn–Cu (Fig.
5-61), Al–Zn (e.g., Simon et al., 1984;
Hoyt and de Fontaine, 1989) or Ni–Si (Po-
lat et al., 1989). Dynamical scaling behav-
ior was also found in some ternary alloys
such as Al–Zn–Mg (Blaschko and Fratzl,
1983). For Cu–Ni–Fe, which was studied
by means of anomalous SAXS (Lyon and
Simon, 1987), the scaling behavior is
found to be obeyed only by the partial
structure functions, indicating that this
system does not behave like a pseudo-bi-
nary system. For Fe–Cr the results are con-
troversial. In contrast to Katano and Iizumi
(1984) and Furusaka et al. (1986), La Salle
and Schwartz (1984) reported that dynami-
cal scaling does nothold. The decomposi-
tion kinetics in Fe–Cr at about 500°C are
fairly sluggish and it may well be that even
after the longest chosen aging time (100 h)
the system had not yet reached the scaling
region where the microstructure displays
self-similarity.
In principle, the structure functionS(
k,
t) contains all the information on the vari-
ous structural parameters of a decomposing
solid such asf(R,t), N
v(t), R
–
(t), morphol-
ogy, etc. which, for instance, may control
its mechanical properties. In practice, how-
ever, commonly only R
–
and N
vand neither
f(R,t) nor the morphology can be extracted
from experimental data. This stems mainly
from a lack of knowledge of the interparti-
cle interference function, which contains
the spatial correlations of the precipitate
microstructure and which manifests itself
in the appearance of a maximum in the S
(
k,t) curves of less dilute systems. Further-
more, both the limited range of 
kover
which S(
k,t) can be measured and the
large background in conventional small-
angle scattering experiments often render
the quantitative extraction of information
on the precipitate microstructure rather dif-
ficult.
From a practical point of view, scaling
analyses are sometimes seen to allow all
information contained in S( k,t) to be deci-
phered. Provided the explicit form of F
˜
(x)
could be predicted on grounds of a first-
Figure 5-61.Time dependence of the scaling func-
tion F˜(
k/k
m)=k
3
m
S(k,t) for Mn–33 at.% Cu at
450°C. For later times (t> 5115 s) F˜(
k/k
m) becomes
time independent and, hence, dynamic scaling holds
(top). The structure functions taken at earlier times
(965, 1602, 2239, 2886 and 3532 s) do not yet dis-
play scaling behavior (bottom). (After Gaulin et al.,
1987.)www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

450. principles theory, it could be compared
with experimental data and employed for a
comparison of S(
k,t) curves taken from
different materials. Up to now, however,
this is not feasible. For this reason various
phenomenological theories for F
˜
(x) have
been conceived (e.g., Furukawa, 1981;
Hennion et al., 1982) amongst which the
model of Rikvold and Gunton (1982) may
be regarded as the one that is most conven-
ient for a comparison with experimental
data as it contains the precipitated volume
fraction as the only parameter. This model
assumes the two-phase microstructure to
consist of a ‘gas’ of spherical second phase
particles (with an identical scattering form
factor) each of which is surrounded by a
zone depleted of solute atoms. With simple
approximations on the probability distribu-
tion for pairs of particles with certain inter-
particle spacings, the explicit analytical
form for F
˜
(x) was derived. However, due
to the various assumptions invoked, the
Rikvold–Gunton model is restricted to
smaller precipitated volume fractions. In
spite of its simplicity, fair agreement was
reported between the theoretical F
˜
(x) and
the scaling functions obtained from com-
puter simulations and from scaling analy-
ses of S(
k,t) curves taken from decom-
posed Al–Zn and Al–Ag–Zn alloys (Simon
et al., 1984). In constrast, in studies of
Al–Zn (Forouhi and de Fontaine, 1987)
and of Ni–Si (Chen et al., 1988) the pre-
dicted F
˜
(x) was found to be much broader
than the experimental ones; scaling analy-
ses for borate glasses also could not verify
the theoretically predicted form of F
˜
(x)
(Craievich et al., 1986). This may stem
from the inadequate assumptions on the
chosen probability distribution in which
long-range correlations are neglected
and/or from precipitate morphologies de-
viating from spheres, e.g., platelets in
Al–Zn.
In contrast to Al–Zn, the small misfit
strains between the Al-rich matrix and the
spherical 
d¢-Al
3Li precipitates renders the
Al–Li system ideal for an accurate test of
scaling in the later stages of coarsening.
Using SAXS, Che et al. (1997) found scal-
ing behavior to be obeyed in the coarsening
regime of each of the Al–Li alloys with dif-
ferent volume fractions ranging from 0.18
to 0.23. The breadth of the scaled structure
function, measured by the full width at half
maximum, versus the equilibrium volume
fraction, agreed well with the boundary in-
tegral-based computer simulation of
Akaiwa and Voorhees (1994) (cf. Sec.
5.6.3).
As a concluding remark to this section it
is probably fair to state that scaling analy-
ses currently can neither furnish the practi-
cal metallurgist with more information on
the precipitate microstructure nor on its dy-
namic evolution than has been possible by
conventional analyses of S(
k,t) curves
prior to the emergence of the scaling hy-
potheses. It is felt that precise information
on the size distribution, the morphology,
and the spatial arrangement of precipitates
in a two-phase microstructure becomes
more readily available from studies em-
ploying direct imaging techniques, e.g.,
CTEM or AFIM, in particular, as dynami-
cal scaling only holds in the later stages of
aging where the precipitate microstructure,
in general, can be easily imaged and re-
solved by these techniques. Furthermore,
once dynamical scaling is satisfied, the
system is close to the asymptotic limit
where the LSW theory or its extensions
may be applied for a prediction of its dy-
namic evolution during further aging.
5.8.2 Power-Law Approximations
So far not explicit assumption has been
made about the time dependence of the
398 5 Homogeneous Second-Phase Precipitationwww.iran-mavad.com 
+ s e  l ≤'4 , kp e  r  i ≤&s ! 9 j+  N   0 e

451. 5.8 Self-Similarity, Dynamical Scaling and Power-Law Approximations 399
chosen characteristic length, e.g., k
m
–1(t),
entering the scaling law, Eq. (5-81). As the
self-similarity of a precipitate microstruc-
ture and, hence, the scaling law is impli-
citly contained in the LSW theory of coars-
ening, the region of validity of dynamical
scaling coincides with the LSW regime of
coarsening. 
k
m
–1(t) is thus expected to show
the simple power-law behavior,
k
m
–1(t) ∫t
a
(5-82)
with a= 1/3. Accordingly, if scaling holds,
the maximum of the structure function
must evolve in time as 
S
m(t) ∫t
b
(5-83)
with b=3a(Eq. (5-81)). Such a power-law
behavior was frequently corroborated by
scattering experiments on materials that
were aged in the scaling region, and also by
computer simulations (e.g., Lebowitz et al.,
1982), which indicated that scaling would
hold.
The more recent theoretical develop-
ments on the kinetics of phase separation
have predicted various other values for the
exponent a. On the basis of their cluster–
diffusion–coagulation model (Sec. 5.6.4),
for intermediate times Binder and cowork-
ers predict a= 1/6 and a= 1/5 or 1/4 for
low and intermediate temperatures, respec-
tively (Binder and Stauffer, 1974; Binder
1977; Binder et al., 1978). As is shown in
Fig. 5-31b, approximation of S
m(t), which
displays some curvature, by a power-law
(Eq. (5-83)) yields b = 0.7 rather than 0.48
as implied by scaling. The LBM theory of
spinodal decomposition, which accounts
for some coarsening at earlierstages (Sec.
5.5.4), yields a= 0.21. As outlined in Sec.
5.5.6, these values agree quite well with
the corresponding values (a= 0.16 to 0.25
and b= 0.41 to 0.71, depending on the
supersaturation and ‘aging temperature’)
obtained from fitting power laws to the
corresponding data from computer simula-
tions.
Inspired by the theoretical predictions,
many scattering experiments on alloys
were interpreted in terms of power-law ap-
proximations. Frequently the existence of
two well-defined kinetic regimes with dis-
tinct values of ahas been reported (e.g.,
Fig. 5-32b). At earlier times, aranges
between Ù0.1 and ≈0.2, whereas at later
stages aand bare found close to the values
1/3 and 1 predicted by the LSW theory.
Sometimes this has been taken as evidence
(e.g., Katano and Iizumi, 1984) for the first
regime to be dominated by the cluster–
diffusion–coagulation mechanism (Sec.
5.6.4), whereas in the second regime, the
evolution proceeds according to the LSW
mechanism via the evaporation and con-
densation of single solute atoms.
However, the interpretation of and S
m(t)
and 
k
m(t) in terms of two distinct kinetic
regimes, each of which is well described by
a power law, seems rather debatable. A
closer examination of the S
m(t) and k
m
curves (e.g., Fig. 5-32b) always reveals
some curvature prior to reaching the scal-
ing region. This clearly shows that the ex-
ponents aand bare time dependent; thus,
apart from the LSW regime, a power-law
approximation must be seen as a rather
poor description of the dynamic evolution
of a decomposing solid and commonly
does not disclose the specific growth
mechanism dominating at a certain aging
regime. This becomes particularly evident
by employing the Numerical Model (Sec.
5.7.3) for a derivation of the exponent a(t)
=∂log R
–
/∂logt. As the N model com-
prises nucleation, growth and coarsening
as concomitant processes on the basis of
just one growth mechanism – single-atom
evaporation or condensation in the LSW
sense – a plot of a(t) versus tallows a
closer examination to be made of the valid-www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

452. ity of power-law approximations at any in-
stant. This is shown in Fig. 5-62 for Cu–1.9
at.% Ti, the experimental kinetic data of
which are well described by the N model
(cf. Figs. 5-53 and 5-55). Towards the end
of the nucleation regime, a(t) increases
sharply from ≈0.15 to its maximum value a
= 0.5
6
which is indicative of a diffusion-
controlled growth of the particles. The du-
400 5 Homogeneous Second-Phase Precipitation
Figure 5-62.Variation of
the time exponent awith ag-
ing time as evaluated for
Cu–1.9 at.% Ti by means of
the N model.
6
Accounting for modified boundary conditions, the
cluster-dynamics approach even yields a maximum
value a = 0.7 (Kampmann et al., 1992).
Figure 5-63.Dynamical scaling of the structure function of Cu-2.9 at.% Ti beyond t= 250 min; SANS resultswww.iran-mavad.com 
+ s e  l ∫'4 , kp e  r  i ∫&s ! 9 j+  N   0 e

453. 5.9 Non-Isothermal Precipitation Reactions 401
ration of the growth regime where R
–
evolves according to the parabolic power
law R
–
∂t
1/2
is rather short for this alloy
(cf. Sec. 5.7.4.3). At the end of the growth
regime, where the particle number density
has reached its maximum value (Fig.
5-55), a(t) drops within about 250 s to a
value of less than 0.1. In the subsequent
transition regime at intermediate times,
a(t) increases continuously and approaches
only slowly the LSW-coarsening regime
where dynamical scaling holds. Evidently,
there is no time regime between the appear-
ance of the growth regime and the LSW re-
gion where power-law behavior is ob-
served. On the other hand, it may be in-
ferred from Fig. 5-62 that the kinetic evo-
lution of R
–
during intermediate aging
stages might be artificially interpreted in
terms of power laws if the time window
covered by the experiment was too short;
in this case any exponent between 0.15 and
0.33 may be derived. Thus, if the time win-
dow for the kinetic experiment is not prop-
erly chosen, a≈0.2 may be obtained, but
without the cluster–diffusion–coagulation
mechanism being operative. As has been
pointed out in Sec. 5.6.4.3, with increasing
supersaturation, the growth regime with
R
–
µt
1/2
disappears completely and a (t)
takes values only between zero and 1/3.
Furthermore, the transition period (where
a< 1/3) becomes shorter. In this case, LSW
coarsening and dynamical scaling are ob-
served after rather short aging times. This is
illustrated in Fig. 5-63 for Cu–2.9 at.% Ti
which satisfies dynamical scaling after ag-
ing for ≈250 min at 350°C, whereas for the
less concentrated Cu–1.9 at.% Ti alloy scal-
ing only holds after about 5¥10
4
min.
5.9 Non-Isothermal Precipitation
Reactions
For many age-hardening commercial al-
loys, the microstructure is established dur-
ing continuous cooling from the processing
temperature, i.e., the solidification or hot-
working temperature in cast or wrought al-
loys, respectively, or during cooling after
laser surface treatments or welding (Brat-
land et al., 1997, and references therein).
The resulting non-isothermal transforma-
tion usually involves nucleation and
growth of the precipitates as concomitant
processes. Modeling of non-isothermal
transformation rates thus ought to account
for the independent variations of the nucle-
ation and the growth rates with temperature
and with the temperature-dependent solu-
bilities in the matrix and the precipitated
phase. Furthermore, heterogeneous nuclea-
tion at lattice defects often must also be
taken into account, as usually the driving
force for homogeneous nucleation be-
comes sufficient only once the alloy has
been cooled sufficiently deep into the mis-
cibility gap.
Whereas previous models, which have
been based on isokinetic behavior and the
additivity concept, are limited to a descrip-
tion of only diffusional growth (or dissolu-
tion during reheating) (e.g., Shercliff 
and Ashby, 1990a, b; Myhr and Grong,
1991a,b; Onsoien et al., 1999), Grong and
Myhr (2000) considered the non-isother-
mal precipitation reaction as a coupled nu-
cleation and growth process in terms of a
numerical solution and two analytical mod-
els. As validated through a comparison
with the numerical results, both analytical
models, i.e., a simplified state variable so-
lution and a solution based on the Avrami
equation, yielded an adequate description
of the overall non-isothermal transforma-
tion behavior comprising the variation ofwww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

454. the nucleation and growth rates as well as
the precipitated volume fraction with tem-
perature and cooling rate.
5.10 Acknowledgements
The authors would like to thank Prof. R.
Bormann and Dr H. Mertins for their criti-
cal comments on this chapter and Mrs E.
Schröder, A. Conrad-Wienands and B.
Feldmann for their kind endurance and as-
sistance in preparing the manuscript. The
support of this work by the Deutsche For-
schungsgemeinschaft (Leibniz-Programm)
is also gratefully acknowledged.
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460. 6 Spinodal Decomposition
Kurt Binder
Institut für Physik, Johannes Gutenberg-Universität Mainz,
Mainz, Federal Republic of Germany
Peter Fratzl
Erich-Schmid-Institut der Österreichischen Akademie der Wissenschaften
und Montan-Universität Leoben, Leoben, Austria
List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 411
6.1 Introduction................................. 414
6.2 General Concepts.............................. 416
6.2.1 Phenomenological Thermodynamics of Binary Mixtures
and the Basic Ideas of Phase Separation Kinetics . . . . . . . . . . . . . . 416
6.2.2 The Cahn–Hilliard–Cook Nonlinear Diffusion Equation . . . . . . . . . . 420
6.2.3 Linearized Theory of Spinodal Decomposition . . . . . . . . . . . . . . . 421
6.2.4 Spinodal Decomposition of Polymer Mixtures . . . . . . . . . . . . . . . 426
6.2.5 Significance of the Spinodal Curve . . . . . . . . . . . . . . . . . . . . . 428
6.2.6 Towards a Nonlinear Theory of Spinodal Decomposition
in Solids and Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
6.2.7 Effects of Finite Quench Rate . . . . . . . . . . . . . . . . . . . . . . . . 441
6.2.8 Interconnected Precipitated Structure Versus Isolated Droplets,
and the Percolation Transition . . . . . . . . . . . . . . . . . . . . . . . . 442
6.2.9 Coarsening and Late Stage Scaling . . . . . . . . . . . . . . . . . . . . . 445
6.2.10 Effects of Coherent Elastic Misfit . . . . . . . . . . . . . . . . . . . . . . 447
6.3 Survey of Experimental Results...................... 450
6.3.1 Metallic Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
6.3.2 Glasses, Ceramics, and Other Solid Materials . . . . . . . . . . . . . . . . 454
6.3.3 Fluid Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
6.3.4 Polymer Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
6.4 Extensions.................................. 460
6.4.1 Systems Near a Tricritical Point . . . . . . . . . . . . . . . . . . . . . . . 460
6.4.2 Spontaneous Growth of Ordered Domains out of Initially Disordered
Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
6.4.3 Phase Separation in Reduced Geometry and near Surfaces . . . . . . . . . 467
6.4.4 Effects of Quenched Impurities; Vacancies; Electrical Resistivity
of Metallic Alloys Undergoing Phase Changes . . . . . . . . . . . . . . . 468
6.4.5 Further Related Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . 470
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

461. 6.5 Discussion.................................. 471
6.6 Acknowledgements............................. 473
6.7 References.................................. 474
410 6 Spinodal Decompositionwww.iran-mavad.com
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462. List of Symbols and Abbreviations 411
List of Symbols and Abbreviations
a lattice spacing
A, B, … symbols for chemical elements
a(t) structural relaxation variable
A,B symbols for Landau expansion coefficients or other constants
a
0,…,a
3 constants
B
ˆ
critical amplitude of the order parameter
c(x),c
i local concentration, concentration of sitei
C
ˆ
critical amplitude of the scattering function
c
s
B
,c
sp(T) concentration at the spinodal curve
c
coex concentration at the coexistence curve
c
–
average concentration
c
crit critical concentration
d spatial dimensionality
D
0 diffusion constant
D
2 damping coefficient of second sound
D
t tracer diffusion coefficient
e electron charge
e
ij elastic strain tensor
e
0
ij
elastic misfit tensor
E
act activation energy
F thermodynamic free energy
D
∫ free energy function
DF* nucleation free-energy barrier
F*
MF mean-field result for the nucleation barrier
f
0 parameter or universal constant of order unity
f
cg free energy density (coarse-grained)
f
D Debye function
g cooling rate
∂ Hamiltonian
ä Planck constant
J concentration current density
J interaction strength of the Ising model
K elastic modulus
k wavevector, scattering vector in diffraction experiments
k modulus ofk
k
B Boltzmann constant
k
c critical wavenumber in Cahn’s theory
k
F radius of Fermi sphere
k
m wavenumber of maximal growth
L length scale of coarse-graining cell
l number of atoms within a cluster
 Liouville operator
M mobilitywww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

463. m
eff effective mass of an electron
N chain length of a polymer
N number of atoms per cm
3
n
L concentration of a cluster containing L atoms
P
L(M) magnetization distribution function
q rescaled wavenumber
Q
B superstructure Bragg spot
r range of effective interaction
r radius of the grain
·R
2
Ò mean-square end-to-end distance of a polymer chain
·R
2
gyr
Ò gyration radius of a polymer chain
R(k) rate factor
r
0 phenomenological coefficient
S(k,t) equal-time structure factor
s(x,t) entropy density
S
˜
rescaled structure factor
T temperature
t time
T
0 temperature at which a quench is started att=0
T
ab Oseen tensor (component)
T
c critical temperature
T
i,T
f initial, final temperature
t
ij elastic stress tensor
T
t tricritical point
u phenomenological coefficient of the Landau expansion
u
2 second sound velocity
V volume
v phenomenological coefficient of the Landau expansion
w elastic energy density
W(l,l¢) cluster reaction matrix
W
A,W
B time constant of element A, B
x Lifshitz–Slyozov exponent
x position vector
x,y,z spatial coordinates
X
A,X
B degree of polymerization of a polymer chain of type A, B
scaling variable
Z-A%, Z-B% atomic numbers of constituents A, B
a phenomenological coupling constant
b critical exponent of the order parameter
g critical exponent of the structure factor
g phenomenological coefficient
G,g rate factors
h viscosity
h
a change of lattice constant with alloy composition
412 6 Spinodal Decompositionwww.iran-mavad.com
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464. List of Symbols and Abbreviations 413
h
T random force
Q temperature at which behavior of polymer in solution is ideal
k factor describing screening of Coulomb interaction
l wavelength
l
c critical wavelength in Cahn’s theory
l
ijmn elastic stiffness tensor
L generalized Onsager coefficient
m chemical potential difference
m elastic modulus
n critical exponent of the correlation length
n
P Poisson coefficient in elasticity theory
x interfacial width; correlation length
xˆ critical amplitude of the correlation length
x
coex correlation length at the coexistence curve
r
1,r
2, … probability distributions
s subunit length of a polymer
s
ij elastic stress tensor
t rescaled coordinate (time)
t time constant
j pair interaction energy
F volume fraction
F
c volume fraction at the critical point
c Flory–Huggins parameter
c
s Flory–Huggins parameter at the spinodal curve
c
n
–1 phenomenological coefficient
y complex order parameter
ABV element-A–element-B–vacancy model
CHC Cahn–Hilliard–Cook (method)
ESA European Space Association
LBM Langer–Baron–Miller (method)
LSW Lifshitz–Slyozov–Wagner (method)
m (index) maximum
MF mean field
p (index) polarization
V vacancy (also as index)www.iran-mavad.com
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465. 6.1 Introduction
This chapter deals with the dynamics of
phase changes in materials, which are
caused by transferring the material into an
initial state that is not thermodynamically
stable (e.g. by rapid cooling (“quenching”)
or, occasionally, rapid heating; in fluids the
system may also be prepared by a rapid
pressure change). It is generally believed
(Gunton et al., 1983; Binder, 1984a, 1987a,
1989; Kostorz, 1991; Wagner and Kamp-
mann, 2000) that inhomogeneousmateri-
als such phase changes may be initiated by
two different types of mechanism, corre-
sponding to two different types of statisti-
cal fluctuations, namely “homophase fluc-
tuations” and “heterophase fluctuations”.
Fig. 6-1 illustrates these fluctuations qual-
itatively for the example of a binary mix-
ture, where the variable to consider is a lo-
cal concentration variablec(x). By “local”
we do not mean the scale of a lattice sitei
(in a crystalline solid), since then the asso-
ciated concentration variablec
icould take
only two values:c
i=1, if the site is taken
by a B atom, andc
i=0,ifitistakenbyan
A atom in an AB mixture. Instead here we
are interested in an averaged concentration
field obtained by “coarse graining” over a
cell of sizeL
3
(orL
2
if we consider a two-
dimensional layer):
(6-1)
wherexis the center of gravity of the cell
over whichc
iis averaged. This intermedi-
ate length scaleLmust be large in compar-
ison with the lattice spacing, since only if
the cell contains many lattice sites is a con-
tinuum description as anticipated in Fig.
6-1 warranted. On the other hand,Lmust
be small in comparison with the length
scales of the statistical fluctuations that we
wish to consider, i.e., the wavelength
lof
the concentration wave in Fig. 6-1a, or the
width
xof the interface between a droplet
of the new phase and its environment in
Fig. 6-1b. As we shall see, the existence of
this intermediate length scaleLseverely re-
cL c
i
i
()x=
cell
−
∑
3
Œ
414 6 Spinodal Decomposition
Figure 6-1.Schematic diagram of unstable thermodynamic fluctuations in the two-phase regime of a binary
mixture AB at a concentrationc
B(a) in the unstable regime inside two branchesc
s
B
of the spinodal curve and (b)
in the metastable regime between the spinodal curvec
s
B
and the coexistence curvec
(1)
coex
. The local concentration
c(r) at a pointr=(x,y,z) in space is schematically plotted against the spatial coordinatexat some timetafter
the quench. In case (a), the concentration variation at three distinct timest
1,t
2, andt
3is indicated. The diame-
ter of the critical droplet, whose cross-section is shown in case (b), is denoted by 2R*, and the width of the inter-
facial region by
x. Note that the concentration profile of the droplet reaches the other branch of the coexistence
curvec
(2)
coex
in the droplet center only for weak “supersaturations” of the mixture, wherec
B–c
(1)
coex
Oc
s
B
–c
Band
R*o
x; for the sake of clarity, the figure is therefore not drawn to scale (Binder, 1981).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

466. 6.1 Introduction 415
stricts the quantitative validity of the theo-
retical concepts developed in the present
chapter (see also Binder, 2000). This chap-
ter will exclusively consider the mecha-
nism of Fig. 6-1a, where in a thermody-
namically unstable initial state long-wave-
length delocalized small-amplitude statis-
tical fluctuations grow spontaneously in
amplitude as the time after the quench in-
creases. For a binary mixture, this mecha-
nism is calledspinodal decomposition
(Cahn, 1961, 1965, 1966, 1968); the rele-
vant fluctuations can then be considered as
a wavepacket of “concentration waves”,
and one such wave is shown in Fig. 6-1a.
As will be discussed below, this mecha-
nism should not occur inside the whole
two-phase coexistence region of the phase
diagram of the mixture, but rather only in-
side a smaller region, the boundary of
which is given by the spinodal curve (Cahn
and Hilliard, 1958, 1959). Between thespi-
nodal curve c
s
B
and the coexistence curve
c
(1)
coex
(or in between the other branch of the
spinodal andc
(2)
coex
the mechanism of drop-
let nucleation and growth is involved (Fig.
6-1a). This latter mechanism is discussed
in the chapter by Wagner et al., 2001).
Apart from these phase transformation
mechanisms triggered by spontaneous ther-
mal fluctuations, heterogeneous nucleation
processes must also be considered (Zettle-
moyer, 1969); near inhomogeneities in
solids such as grain boundaries, disloca-
tions, external surfaces, or point defects
such as substitutional or interstitial impur-
ities or clusters thereof, microdomains of
the new phase may already be formed be-
fore the quench, or at least their formation
after the quench is greatly facilitated. Gen-
eral theoretical statements about these
heterogeneous mechanisms, however, are
hardly possible without a detailed discus-
sion of the nature of such defects. Such
problems will not be discussed in this
chapter, and the spontaneous growth of
thermal fluctuations dominating phase
changes of unstable but nearly ideal (i.e.,
defect-free) systems will be emphasized.
Although most of the discussion refers to
binary mixtures, an extension to ternary
mixtures is often possible.
Unmixing of binary or ternary mixtures
is not the only phase change where sponta-
neous growth of statistical fluctuations oc-
curs. Consider, for example, an alloy A–B,
which in thermal equilibrium undergoes an
order–disorder phase transition form a dis-
ordered state, where the two species of at-
oms A and B are distributed at random over
the available lattice sites, to an ordered ar-
rangement. There A and B atoms preferen-
tially occupy sites on sublattices (e.g.,b-
CuZn, Cu
3Au, CuAu, Fe
3Al, and FeAl). If
we quench such an alloy from the disor-
dered regime to a state which in equilib-
rium should be ordered, the unstable disor-
dered initial state may also decay by spon-
taneous growth of fluctuations. The dis-
tinction from Fig. 6-1a is that the wave-
length of the growing concentration wave
is not large but coincides with the lattice
spacing of the superstructure. Neverthe-
less, the theoretical treatment of thisspino-
dal ordering(De Fontaine, 1979) is similar
to the theory of spinodal decomposition,
and will also be discussed in this chapter,
which emphasizes the theoretical aspects
(see Wagner and Kampmann (2000) for a
complementary treatment emphasizing the
point of view of the experimentalist).
One of the basic questions in the applica-
tions of materials is to understand the pre-
cipitated microstructure that forms in the
late stages of a phase separation process.
One of the main characteristics of this mi-
crostructure concerns its morphology: the
minority phase may either form compact
islands well separated from each other or it
may form an irregular interconnected net-www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

467. work-like structure (Fig. 6-2). Obviously, a
critical volume fraction exists where this
interconnected net first appears, a so-called
percolation threshold(Stauffer, 1985). Of-
ten it is assumed that the morphology of
well separated islands must have formed
by nucleation and growth, while the perco-
lating morphology is taken as the signature
of the spinodal curve. However, such an
identification is misleading, as will be dis-
cussed in detail in this chapter: the spinodal
curve at best has a meaning for the initial
homogeneous state and controls the initial
stages of phase separation; in the late
stages, however, a generally universal
coarsening behavior occurs (which is sen-
sitively affected by defects!), irrespective
of whether the morphology is intercon-
nected or not. Since a closely related uni-
versal coarsening behavior also occurs in
fluid mixtures and can be studied fairly
well in these systems, we include a discus-
sion of phase separation in fluid mixtures
throughout this chapter. Moreover, “poly-
mer alloys” can be conveniently produced
by quenching from the fluid phase of poly-
mer mixtures, and have become a practi-
cally important class of materials (Hashi-
moto, 1993). 6.2 General Concepts
6.2.1 Phenomenological Thermodynamics
of Binary Mixtures and the Basic Ideas
of Phase Separation Kinetics
We first consider the dynamics of phase
transformations induced by an instantane-
ous quench from the initial temperatureT
0
to a final temperatureT(the idealization of
this infinitely rapid cooling will be relaxed
in Sec. 6.2.7). The simplest case where
such a sudden change of external parame-
ters causes a phase transition is a system
undergoing an order–disorder phase tran-
sition at a critical temperatureT
cwith
T<T
c<T
0(Fig. 6-3a). The initially disor-
dered system is then unstable, and immedi-
ately, small ordered domains form. As the
timetafter the quench passes, the order pa-
rameter in these domains must reach its equi-
librium value (±
y), and the domain size
must ultimately grow to macroscopic size.
Alternatively, we consider a solid or liq-
uid mixture A–B with a miscibility gap
(Fig. 6-3b). Quenching the system at time
t= 0 from an equilibrium state in the single
phase region to a state underneath the co-
existence curve leads to phase separation;
in thermal equilibrium, macroscopic re-
gions of both phases with concentrations
c
(1)
coex
andc
(2)
coex
coexist.
416 6 Spinodal Decomposition
Figure 6-2.Snapshot pic-
tures of a two-dimensional
system undergoing phase
separation with a volume
fraction of (a)
F= 0.21 and
(b)
F= 0.5. The pictures
were obtained by numerical
solution of the non-linear
Cahn–Hilliard equation, Eq.
(6-14) (Rogers and Desai,
1989).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

468. 6.2 General Concepts 417
Of course, phase transformations com-
bining an ordering process (such as in Fig.
6-3a) with an unmixing process (such as in
Fig. 6-3b) also occur; as we are interested
mainly in the basic concepts of these pro-
cesses, we will not consider these more
complex phenomena here, but will return
to this problem in Sec. 6.4.1.
The basic concepts of the kinetic mech-
anisms of the processes indicated in Fig.
6-3a and b invoke the idea of extending the
concept of the thermodynamic Helmholtz
free energyFto states out of equilibrium
(Gunton et al., 1983; Binder, 1987a). In
the single-phase region of the A-rich alloy,
Ffirst decreases with increasing concen-
tration of B atomsc
B, owing to the entropy
of mixing. Similarly, in the B-rich single-
phase region,Fdecreases with increasing
concentration of A atoms,c
A=1–c
B.Inthe
two-phase coexistence region in equilib-
rium, we have a linear variation ofFwith
composition; this reflects the fact that the
amounts of the two coexisting phases (with
concentrationsc
(1)
coex
,c
(2)
coex
) change linearly
withc
Baccording to the lever rule.
Mean-field type theories (Gunton et al.,
1983; Binder, 1987a) suggest that a free
energyF¢ÙFcan also be introduced de-
scribing single-phase states within the two-
phase region. SinceF¢must coincide with
Fforc
(1)
coex
andc
(2)
coex
, inevitably a double-
well structure results, as indicated by the
dash-dotted curve in Fig. 6-3c. This hypo-
thetical free energyF¢now allows a further
distinction to be made: states where
(∂
2
F¢/∂c
2
B
)
T> 0 are calledmetastable,and
states where (∂
2
F¢/∂c
2
B
)
T< 0 are calledun-
stable. The locus of inflection points in the
(T,c
B) plane,
(∂
2
F¢/∂c
2
B
)
T= 0 (6-2)
Figure 6-3.(a) Order parameter yof a second-order phase transition vs. temperature, assuming a two-fold
degeneracy of the ordered state (described by the plus and minus signs of the order parameter). The quenching
experiment is indicated. (b) Phase diagram of a binary mixture with a miscibility gap ending in a critical point
(T
c,c
B
crit) of unmixing, in the temperature–concentration plane. Again the quenching experiment is indicated,
and the quenching distances from the coexistence curve (dT) and from the critical point (DT) are indicated.
(c) Free energy plotted versus composition at temperatureT(schematic). For further explanations, see text
(Binder, 1981).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

469. defines the spinodal curvec=c
s
B
(T). This
distinction is now linked to the two trans-
formation mechanisms shown in Fig. 6-1;
it is the unstable regime inside the two
branches of the spinodal curve, where
long-wavelength fluctuations spontane-
ously grow rather than decay. At a later
stage, the inhomogeneous concentration
distribution thus generated will coarsen. In
the metastable regime, the system is stable
against such weak (small-amplitude) fluc-
tuations, and localized large-amplitude
fluctuations (“droplets” of the new phase)
must form in order to start the transforma-
tion.
This idea implies, as will be explained
below, asingular transition in the kinetic
transformation mechanismat the spinodal
curve. However, this spinodal singularity is
really only an artefact of an over-simplified
theoretical picture: apart from the very
special limit of infinitely weak, infinitely
long-ranged forces in which mean-field
theory becomes correct (Penrose and Lebo-
witz, 1971; Lebowitz and Penrose, 1966;
Binder, 1984b), the transition from the nu-
cleation mechanism to the spinodal decom-
position mechanism is completely gradual
(Binder et al., 1978). As will be discussed
in more detail in Sec. 6.2.5, the spinodal
curve cannot be unambiguously defined.
At this point, we recall that the “weak delo-
calized long-wavelength fluctuations” can-
not be identified in terms of the atomic
concentration variablec
i, which undergoes
rapid large-amplitude (c
i=0toc
i=1!) vari-
ations from one lattice site to the next, but
implies the introduction of the coarse-
grained concentration fieldc(r), Eq. (6-1).
The free energyF¢of “homogeneous”
states in the two-phase region depends on
the length scaleLover which short-wave-
length concentration fluctuations have
been integrated (Fig. 6-4). This “coarse-
grained” free-energy densityf
cgis not pre-
cisely identical to the true free-energy den-
sityfin the single-phase region, since con-
centration variations with wavelengths ex-
ceedingLcontribute tofbut are excluded
fromf
cg. However, this difference is minor,
and in the single-phase region we may con-
sider the limitLÆ∞and thenf
cgtends to-
wardsfuniformly. This is not possible in
the two-phase region, however, wheref
cg
describes homogeneous states only if
LÁ
x, the interfacial width; ifLÔ xthen
418 6 Spinodal Decomposition
Figure 6-4.Schematic plot of
the coarse-grained free-energy
densityf
cg(f) as a function of the
order parameter
f=(c–c
crit)/c
crit
in a first-order transition from
f
1
coextof
2
coex, for a “symmetric”
situation withf(
f
1
coex)=f(f
2
coex)
as in the Ising model. Spinodals
f
s
(1, 2)(L) defined from inflection
points off
cg(f) depend distinctly
on the coarse-graining lengthL
and the interfacial width
x(Bin-
der, 1987a).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

470. 6.2 General Concepts 419
the states that yield dominating contribu-
tions tof
cgare phase separated on a local
scale, and thereforef
cgtends smoothly to-
wards the double-tangent construction as
LÆ∞. Hence there is no unique theoreti-
cal method of calculating a spinodal curve;
remember that mean-field theories, which
yield spinodal curves easily, are only inac-
curate descriptions of real systems. Simi-
larly, extrapolation procedures by which
spinodal curves are extracted from experi-
mental data also involve related ambigui-
ties, as will discussed in Sec. 6.2.5.
Hence the coarse-graining alluded to
above means that a microscopic Hamil-
tonian
∂{c
i} of the binary mixture is re-
placed by a so-called free-energy function
D
∫{c(x)}. For example, the Hamiltonian
may correspond to a standard Ising-type
pairwise interaction model (Binder, 1986;
De Fontaine, 1979):
where
jis the interaction energy between a
pair of atoms.
Carrying out the coarse-graining defined
in Eq. (6-1) we expect that
∂{c
i}/k
BTwill
be replaced by
wheredis the spatial dimensionality of the
system andris the range of the effective
interaction in Eq. (6-3) (D
j(x
i–x
j)∫j
AA
+j
BB–2j
AB):
(6-5)
r
d
j
ij ij
j
ij
2
2
2
=
∑
∑−−
−
()()
()
xx xx
xx
D
D j
j
D∫{( )}
{ [ ( )]/ [ ( )] }
c
kT
xf c kT r c
d
x
xx
B
cg B
(6-4)
=d +
∫ ∇
1
2
22
∂{{()
()[()()]
()()()}
cc c
cccc
cc
i
ij
ijij
iji j j i
ij i j} = (6-3)
BB
AB
AA
1
2
11
11
≠
∑ −
+− −+−
+−−−j
j
jxx
xx
xx
The term
1
–
2
r
2
[—c]
2
in Eq. (6-4) accounts
for the free energy cost of inhomogeneous
concentration distributions. Here the
coarse-grained free-energy densityf
cg(c)
resulting from Eqs. (6-3) and (6-1) by car-
rying out the restricted trace over the{c
i}
for a fixed concentration field {c(x)} is
very difficult to obtain in practice; qualita-
tively the behavior off
cg(c) should be sim-
ilar to the mean-field (MF) result for the
free energy of a binary mixture,
(6-6)
with
(6-7)
NearT
c
MFwe can replacef
MF(c) by its Lan-
dau expansion,
(6-8)
whereA~(T/T
c
MF– 1) < 0 forT<T
c
MFand
B> 0. It is well known that the actual
critical temperatureT
cdoes not coincide
with the mean-field prediction. Therefore,
it is usually assumed that the parameters
appearing in the actual coarse-grained
free energyf
cg(c) are not the mean-field
parametersA,B,…, but rather these pa-
rameters are “renormalized” due to short-
wavelength fluctuations, and alsor(Eq.
(6-5)) may thus be modified. Therefore,
f
cg(c)andr are not calculated from micro-
scopic models such as Eq. (6-3), but are
treated as phenomenological input parame-
ters of the theory, which are fitted to ex-
periment.
fcfAcc
Bc c
MF crit
crit=() ( )
()
0
2
4+−
+− +…
kT
j
ijBc
MF
≡−∑Dj()xx
1
11
21
kT
fccc c c
T
T
cc
B
MF
c
MF
=( ) ln ( ) ln( )
()
+− −
+−www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

471. 6.2.2 The Cahn–Hilliard–Cook
Nonlinear Diffusion Equation
Since in the total volumeVthe average
concentration,
c
–
= (1/V )Údxc(x,t) (6-9)
is conserved, the time-depent concentra-
tion fieldc(x,t) satisfies a continuity equa-
tion,
(6-10)
wherej(x,t) is the concentration current
density. Following standard nonequilib-
rium thermodynamics (de Groot and Ma-
zur, 1962),j(x,t) is assumed to be propor-
tional to the gradient of the local chemical
potential difference
m(x,t):
j(x,t)=–M —
m(x,t) (6-11)
whereMis a mobility that is discussed be-
low.
In the thermal equilibrium the chemical
potential difference is given as a partial de-
rivative of the Helmholtz energyF(c,T):
m=(∂F/∂c)
T; (6-12)
remember that the condition for two-phase
coexistence, the equality of chemical po-
tential differences
is the physical content of the double-tan-
gent construction shown in Fig. 6-3c. Eq.
(6-12) is generalized to an inhomogeneous
nonstationary situation far from equilib-
rium, where bothc(x,t)and
m(x,t) depend
on space and time, by defining
m(x,t)as
a functional derivative of the Helmholtz
energy functionalD
∫in Eq. (6-4):
m(x,t)∫d(D ∫{c(x,t)})/dc(x,t) (6-13)
Inserting Eq. (6-14) into Eq. (6-13) yields
m(x,t)=(∂f
cg/∂c)
T–r
2
k
BT—
2
c(x,t)
mm
12
12===
coex coex
(/) (/)() ( )∂∂ ∂∂Fc Fc
Tc Tc||
∂
∂
+∇⋅
ct
t
t
(,)
(,)
x
jx=0
and using this result in the continuity rela- tion, Eq. (6-10), we obtain the Cahn–Hil- liard nonlinear diffusion equation (Cahn, 1961):
One immediately obvious defect of this
equation is its completely deterministic character, which implies that random sta- tistical fluctuations are disregarded (apart from fluctuations included in the initial condition, the state at temperatureT
0where
the quench starts). This defect can be rem- edied, following Cook (1970), by adding a random force term
h
T(x,t) to Eq. (6-14):
Here
h
T(x,t) is assumed to be delta-corre-
lated Gaussian noise, and the mean-square amplitude·
h
2
T
Ò
Tis then linked to the mobil-
ityMvia a fluctuation-dissipation relation.
·
h
T(x,t)h
T(x¢,t¢)Ò
T
=·h
2
T
Ò
T—
2
d(x–x¢)d(t–t¢) (6-16)
·
h
2
T
Ò
T=2k
BTM (6-17)
Eqs. (6-15)–(6-17) constitute the main re- sults of this section, on which all further treatment presented here is based. At this point we stress the main assumptions that have been made either explicitly or tacitly:
(i) Effects due to the lattice anisotropy of
the solid have been ignored. In a crystalline solid the interfacial free energy between co-existing A-rich and B-rich phases will
∂
∂
∇
×
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟−∇
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
+
ct
t
M
fc t
c
rkT c t
t
T
T
(,)
[(,)]
(,)
(,)
x
x
x
x
= (6-15)
cg
B
2
22
h
∂
∂
∇
×
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟−∇
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
ct
t
M
fc t
c
rkT c t
T
(,)
[(,)]
(,)
x
x
x
= (6-14)
cg
B
2
22
420 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

472. 6.2 General Concepts 421
depend on the orientation of the interface;
only in the limitTÆT
cdoes this interfa-
cial free energy become isotropic (Wortis,
1985). The isotropic form of Eq. (6-15)
outside the critical region would only hold
for phase separation in isotropic amor-
phous solids.
(ii) It is assumed that the local concen-
tration field is the only slowly relaxing var-
iable whose dynamics must be considered
explicitly after the quench, while all other
variables equilibrate instantaneously. This
assumption is not true, of course, if phase-
separating fluids are considered where the
coupling of hydrodynamic variables and
the resulting long-range hydrodynamic
interactions need to be taken into account
(Kawasaki and Ohta, 1978). However, in
solids other slow variables may also occur,
particularly near the glass transition (in
amorphous solids) or near other phase
transformations. A treatment of spinodal
decomposition in the presence of a cou-
pling to slowly relaxing structural vari-
ables is only possible in very simplified
cases (Binder et al., 1986).
(iii) It is assumed that the spatial concen-
tration variations of interest are small, i.e.,
in Eq. (6-4) terms of the order [—
2
c(x)]
2
(and higher) can be neglected in compari-
son with the lowest-order gradient term,
[—c(x)]
2
. This assumption is true near the
critical point of unmixing, where the inter-
facial width
xis much larger than the inter-
atomic spacing, but it is not true in the gen-
eral.
(iv) The concentration dependence of the
mobilityMis neglected in the derivation
yielding Eqs. (6-14) and (6-15). Again, this
approximation is valid nearT
c, since then
the relevant scale for concentration varia-
tions becomes small ((c
(2)
coex
–c
(1)
coex
)/c
critO1),
but is not true in general.
(v) The use of a continuum description
for a solid (Eqs. (6-1) and (6-4)) requires
that a coarse-graining is performed over a
length scale much larger than the lattice
spacing, but less than
x, which again re-
stricts the region of validity of the theory to
the critical region. On the other hand, fluc-
tuations and nonlinear phenomena are
known to lead to a breakdown of meanfield
theory near the critical point, and nontrivial
critical phenomena arise (Stanley, 1971;
Binder, 2001). As we shall see (Secs. 6.2.5
and 6.2.6) this also restricts the applicabil-
ity of the theory.
6.2.3 Linearized Theory
of Spinodal Decomposition
It may seem unsatisfactory that the basic
equation of the theory (Eq. (6-15)) is sup-
posed to hold only under fairly restrictive
conditions, but what is even worse is the
fact that this equation completely with-
stands an analytical solution, and brute
force numerical approaches where Eq. (6-
15) is attacked by large-scale computer
simulation are needed (Meakin et al., 1983;
Petschek and Metiu, 1985; Milchev et al.,
1988; Gawlinski et al., 1989; Rogers et al.,
1988; Toral et al., 1988; Oono and Puri,
1988; Puri and Oono, 1988). We shall re-
turn to these approaches in Sec. 6.2.6, and
also address the complementary approach
of Monte Carlo simulations based directly
on the microscopic Hamiltonian, Eq. (6-3)
(Bortz et al., 1974; Marro et al., 1975,
1979; Rao et al., 1976; Sur et al., 1977;
Binder et al., 1979; Lebowitz et al., 1982;
Fratzl et al., 1983; Heermann, 1984a, b,
1985; Amar et al., 1988). Here we prefer to
discuss theassumptionthat in the initial
stages of unmixing the fluctuation
dc(x,t)∫c(x,t)–c
–
is small everywhere in the system. SinceL
cannot be made arbitrarily large,this as-
sumption is not typically true, as we shallwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

473. see below (Secs. 6.2.5 and 6.2.6), but nev-
ertheless it is instructive to study it!
Given this assumption, we may linearize
Eq. (6-15) (or Eq. (6-14) if we also neglect
the thermal noise) indc(x,t). Then Eq. (6-
14) becomes
and, introducing Fourier transformations
dc
k(t)∫∫d
d
xexp(ik·x)dc(x,t) (6-19)
Eq. (6-18) is solved by a simple exponen-
tial relaxation,
dc
k(t)∫dc
k(0) exp[R (k)t] (6-20)
with the rate factorR(k):
(6-21)
R(k)∫Mk
2
[(∂
2
f
cg/∂c
2
)
T,c
–+r
2
k
BTk
2
]
The equal-time structure factorS(k,t)at
timetafter the quench:
S(k,t)∫·dc
–k(t)dc
k(t)Ò
T (6-22)
where·…Ò
Tdenotes a thermal average,
then also exhibits a simple exponential re-
laxation:
S(k,t)∫S
T0
(k)exp[2R(k)t] (6-23)
Here the prefactor
(6-24)
S
T0
(k)∫·dc
–k(0)dc
k(0)Ò
T∫·dc
–kdc
kÒ
T0
is simply the equal-time structure factor in
thermal equilibrium at temperatureT
0be-
fore the quench. Note thatR(k) is positive
for 0 <k<k
c, with
(6-25)
k
c∫2p/l
c=[–(∂
2
f
cg/∂c
2
)
T,c
–/(r
2
k
BT)]
1/2
Thus, whereas the structure factor should
exhibit exponential growth in this region,
fork=k
cit should be time-independent,
S(k
c,t)=S(k
c, 0). However, neither in ex-
∂
∂
∇
×
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
−∇
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
t
ctM
fc
c
rkT c t
Tc
d
d
(,)
()
(,)
,
x
x
= (6-18)
cg
B
2
2
2
22
periment (Fig. 6-5a, b) nor in simulations
(Figs. 6-5c, d) is such a time-independent
intersection point observed. Also in the
Cahn plot,R(k)/k
2
plotted versusk
2
, in-
stead of the predicted linear behavior
(Fig. 6-6a), 2R(k)/k
2
=2D
0(1 –k
2
/k
c
2) with
a negative diffusion constant
D
0=–M(∂
2
f
cg/∂c
2
)
T,c
–(uphill diffusion)
curvature is typically observed (Fig. 6-6b).
There are several possible reasons why
the simple linearized theory of spinodal de-
composition as it has been outlined so far is
invalid:
(i) Fluctuations in the final state at the
temperatureTmust be included (Cook,
1970). This problem will be considered at
the end of this subsection.
(ii) Nonlinear effects are important dur-
ing the early stages of the quench. This
problem will be discussed in Secs. 6.2.5
and 6.2.6.
(iii) There is already an appreciable re-
laxation of the structure factor occurring
during the quench fromT
0toTif the cool-
ing rateg=–dT/dtis finite (see Sec. 6.2.7).
(iv) The concentration field is coupled to
another slowly relaxing variable (Binder et
al., 1986; Jäckle and Pieroth, 1988). Here
we only very briefly outline the idea of the
approach of Binder et al. (1986). Accord-
ing to Eqs. (6-23) and (6-21), the maximum
growth rate of the structure factor occurs at
R
m=R(k
m),k
m=k
c/÷
–
2 (6-26)
Suppose now the concentration couples
linearly to a non-conserved variablea(t)
describing, for example, structural relaxa-
tion, whose fluctuations in the absence of
any coupling would decay exponentially,
proportional to exp(–
gt). The decay of
concentration fluctuations will be affected
(i) if the coupling between the variables
a(t)andc(x,t) is sufficiently strong, and
422 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

474. 6.2 General Concepts 423
Figure 6-5.(a) Neutron small-angle scattering intensity vs. scattering vectork( b: scattering angle) for an
Au–60 at.% Pt alloy quenched to 550 °C (Singhal et al., 1978). (b) Time evolution of the structure factor
at 541 K in Al–38 at.% Zn after subtraction of the prequench scattering. Solid lines are the best fit to the data
using the LBM theory, Eq. (6-60) (Mainville et al., 1997). (c) Time evolution of the structure factorS(k,t)ac-
cording to a Monte Carlo simulation of a three-dimensional nearest-neighbor Ising model of an alloy at critical
concentration and temperatureT=0.6T
c. Due to the periodic boundary condition for the 30¥30¥30 lattice,kis
only defined for discrete multiples of (2p)/30; these discrete values ofS(k,t) are connected by straight lines
(Marro et al., 1975). (d) Time evolution of the normalized structure factorS
˜
(k,t)vs.kfor the discrete version
of the Ginzburg–Landau model (Eq. (6-4)), namely
wherec
lis a continuous variable representing the average concentration in thelth cell of sizeL¥L, and the
phenomenological constantsA,B,andC have been chosen asA/C= – 2.292,÷
-
B/C= 0.972 (Ccan be scaled out
by redefining thec
ls). Data are for anN=40¥ 40 lattice with periodic boundary conditions at timest= 0, 10,
20, …, 90 Monte Carlo steps (MCS) per site. The arrow indicates the estimate for the wavenumber of maximum
growth,k
m(0) =k
c/÷
-
2=÷
----
–A/C(Milchev et al., 1988).
∂/ [()()] ()kT Ac c Bc c Cc c
l
ll
lm
lB crit crit m
=∑∑ −+− + −
〈〉
24 1
2
2www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

475. (ii) if the ratesR
mandgare of the same
order. In the absence of any coupling
between these variables their fluctuations
would decay independently of each other,
with decay rates being given as
G
+(k)=g
(relaxation ofa(t) is then independent ofk)
and
G
–(k)=–|D
0|k
2
(1 –k
2
/k
c
2) (spinodal
decomposition). Thismode spectrumof
uncoupled structural and concentration
fluctuations is shown by the dashed curves
in Fig. 6-7). However, both relaxation rates
become strongly modified if these vari-
ables are coupled. The strength of this
coupling can be related to 1–D
∞/D
0, where
D
0andD
∞are the low- and high-frequency
limits of the diffusion constant. Full curves
in Fig. 6-7 show for three (arbitrary)
choices of parameters, the relaxation rates
G
+(k) andG
–(k) for the case of coupled
variables. It is seen that a “mixing” of
interdiffusion and relaxation occurs; for
smallkthe interdiffusion (or spinodal de-
composition) is given by
G
–(k), for largek
by
G
+(k), and for intermediatekboth expo-
nentials exp[–
G
+(k)t)] and exp[–G
–(k)t)]
contribute to the growth and/or decay of
concentration fluctuations. Plotting the
mode
G
–(k) which describes spinodal de-
composition for smallkin the form of a
Cahn plot, we encounter pronounced cur-
vature.
Such a coupling whereR
mandgare of
comparable size might occur if we study
spinodal decomposition in glasses (see for
example Yokota, 1978; Acuña and Craie-
vich, 1979; Craievich and Olivieri, 1981)
or in fluid polymer mixtures near their
glass tansition (e.g., Meier and Strobl,
1988). The slow variables are then ex-
pected to relax with a broad spectrum of
rates rather than with a single rate
g. Fluc-
424 6 Spinodal Decomposition
Figure 6-6.Schematic “Cahn plot”,
i.e.,R(k)/k
2
vs.k
2
, (a) as predicted by
the linear theory and (b) as it is typi-
cally observed. Ideally, the “Cahn
plot” should yield a straight line: the
intercept with the abscissa occurs at
k
c
2, and the intercept with the ordinate
atD
0(Binder et al., 1986).
Figure 6-7.Mode spectrum { G
+(k),G
–(k)} (full
curves) of an unmixing system coupled to a slow var- iable plotted vs.k
2
/k
c
2for three parameter choices. In
the absence of any coupling, the slow variable would relax with a rate
G
+(k)=g(broken horizontal straight
lines) and the unmixing system would relax with a rate
G
–(k)=4R
m(k
2
/k
c
2)(1–k
2
/k
c
2) (broken curves).
All rates are normalized byR
m(Binder et al., 1986).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

476. 6.2 General Concepts 425
tuations in the final state also need to be in-
cluded in this case (Jäckle and Pieroth,
1988).
We now discuss the effect of fluctuations
in the final state for the simplest case, Eq.
(6-15), treated in the framework of the lin-
earization approximation. After some sim-
ple algebra we obtain from Eqs. (6-15) to
(6-22) the following equation (Cook,
1970):
It is seen that an effective diffusion con-
stant for uphill diffusion, defined as
(6-28)
Dkt
kt
Skt
eff
d
d
(,) ln (,)≡
1
2
2
d
d
=(6-2)
cg
BB
t
Skt Mk
f
c
rkTk Skt kT
Tc
(,)
(,)
,
−
×
∂
∂
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
−
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
27
2
2
2
22
now yields
(6-29)
Since in the linear theoryS(k,tÆ∞)Æ∞,
D
eff(k,tÆ∞) reduces to the simple Cahn
(1961) result quoted above, but this limit is
never reached owing to nonlinear effects.
On the other hand, att= 0, Eq. (6-29) leads
to a linear relation betweenD
eff(k, 0) and
k
2
again, because
(6-30)
is linear ink
2
also. Here the point where
D
eff(k, 0) changes sign is notk
c, but shifts
to a larger value. This behavior is illus-
trated in the upper part of Fig. 6-8 where
the (rescaled) structure factorS
˜
(q,
t) and
(rescaled) effective diffusion constant
[(,)]
,
Sk
f
c
rkTk kT
Tc
0
1
2
2
22
0
0
−
∂
∂
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
cg
BB
Dkt D
k
k
kTMSkt
eff
c
B(,) / (,)≡− −
⎛
⎝
⎜
⎞
⎠
⎟+
0
2
21
Figure 6-8.Scaled structure
function (left) and normalized
diffusion constantD
˜
eff(q,t)∫
q
2
d[lnS
˜
(q, t)]/dt(right) vs.q
orq
2
, respectively, for an instan-
taneous quench from infinite
temperature toT/T
c=4/9. Top,
Cahn–Hilliard–Cook (CHC)
approximation, Eqs. (6-27) and
(6-29), bottom based on the
Langer–Baron–Miller (LBM)
(1975) theory. Ten times
t=1,
2, …, 10 are shown after the
quench. Note that in the CHC
approximation we use the
renormalized value
m
–(Fig.
6-18) of (∂
2
f
cg/∂c
2
)
T,c
–instead
(∂
2
f
cg/∂c
2
)
T,c
–,m
–(T,t= 0) = 0.65
in the present units (Carmesin
et al., 1986).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

477. D
˜
eff(q,t) following from Eqs. (6-27) and
(6-29) are plotted against the (rescaled)
square of the scattering vector,q
2
(where
q=k/k
c,t=2Mr
2
Tk
c
4t,S
˜
=r
2
k
c
2S). It is
seen that on the level of the Cahn–Hilli-
ard–Cook (CHC) approximation there is
initially some shift in the position at which
S(k,t) has its maximum, although this shift
is more pronounced when nonlinear effects
are taken into account, as is approximated
by the theory of Langer et al. (1975), which
we shall refer to as the Langer–Baron–
Miller (LBM) approximation. It is seen that
the main distinction between the CHC and
LBM approximations during the early stages
is the lack of a common intersection point in
the LBM approximation. Also the growth of
S
˜
(q,
t) withtis generally slower (note the
difference in the ordinate scales!). Qualita-
tively, however, the behavior is similar, and
this is also true for the behavior ofD
˜
eff(q,t).
Note thatD
˜
eff(q,t) is distinctly curved in
both cases (apart from the limit
tÆ0).
6.2.4 Spinodal Decomposition
of Polymer Mixtures
Here we consider the modifications nec-
essary for describing binary mixtures of
long flexible macromolecules. In a dense
polymer melt, the configurations of these
linear chain molecules (we disregard here
star polymers, branched polymers, copol-
ymers, and also interconnected networks,
although some aspects of the theory can be
extended to these more complex situations,
see Binder and Frisch (1984)) are random
Gaussian coils interpenetrating each other.
In terms of subunits (“Kuhn segments”) of
lengths
s
Aands
Bfor the two types of
polymers A and B, the degree of polymer-
izationX
AandX
Bis expressed in terms of
chain lengths N
AandN
BasX
A=N
Ag
Aand
X
B=N
Bg
B, if each subunit containsg
Aand
g
Bmonomers. The mean square end-to-end
distance and gyration radius of these coils
then is (Flory, 1953; De Gennes, 1979):
·R
A
2Ò=s
2
A
N
A,·R
B
2Ò=s
2
B
N
B, (6-31)
·R
2
gyr,A
Ò=
1
–
6
s
2
A
N
A,·R
2
gyr,B
Ò=
1
–
6
s
2
B
N
B
In the Flory–Huggins (FH) approximation
(Flory, 1953; Koningsveld et al., 1987) the
expression corresponding to Eq. (6-4) be-
comes (De Gennes, 1980)
(6-32)
where
F(x) is the volume fraction of A seg-
ments, 1 –
F(x) is the volume fraction of B
segments, the mixture is assumed to be
incompressible, and thelattice spacing aof
the Flory–Huggins lattice model is given
bya
2
/[F(1–F)]∫s
2
A
/F+s
2
B
/(1–F). Note
that unlike Eq. (6-4) the parameterais not
related to the range of interaction, but
rather reflects the random coil structure of
the polymer chains. Owing to the connec-
tivity of the polymer chains, the entropy of
mixing term is much smaller than in Eq. (6-
6) (Flory, 1593):
where the Flory–Huggins parameter
ccon-
tains all enthalpic contributions that lead to
unmixing. If
cdoes not depend on volume
fraction, then the spinodal curve resulting
from Eq. (6-33) is given by
2
c
s(F)=(FN
A)
–1
+[(1–F)N
B]
–1
(6-34)
and the critical point occurs at
F
cAB
c A B=
=
(/ )
()/
//
NN
NN
+
+
−
−−
1
2
1
12 12 2
c
1
11
1
kT
f
N
N
B
FH
A
B
= (6-33)()
ln
()ln()
()F
FF
FF
FF
+
−−
+− c
D∫{()}
[ ( )]/
()
[()]F
F
FF
Fx
x
x
kT
xf kT
a
d
B
FH B
=d∫{
+
−
∇
⎫
⎬
⎭
1
2
1
1
22
426 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

478. 6.2 General Concepts 427
These results assume well-defined chain
lengths (“monodisperse polymers”), but
can be generalized to a distribution
of chain lengths (“polydispersity”; see
Joanny, 1978).
Another modification concerns the con-
stitutive kinetic equation: instead of Eq. (6-
11) the connectivity of the chains requires
a non-local relationship for the current den-
sityJ
F(x,t) relating to volume fractionF
(De Gennes, 1980; Pincus, 1981),
J
F(x,t)=–Ú L(x–x¢)— m(x¢,t)dx¢(6-35)
where
L(x–x¢) is a generalized Onsager
coefficient describing polymer–polymer
interdiffusion. The rateR(k) in Eq. (6-21)
is then expressed in terms of the Fourier
transform
L
kofL(x–x¢), i.e.,
R(k)=k
2
L
k[S
T
coll(k)]
–1
(6-36)
where the expression for the effective col-
lective structure factorS
T
coll(k) can be ex-
pressed in terms of the structure factors
S
A(k) andS
B(k) of the single chains as
(Binder, 1983, 1984c, 1987b)
[S
T
coll(k)]
–1
=[FS
A(k)]
–1
+[(1–F)S
B(k)]
–1
–2c
eff
(F,T,k) (6-37)
where 2
c
eff
(F,T,k) is a wavevector-
dependent generalization of the term re-
sulting from the second derivative of
∂
2
[cF(1 –F)]/∂f
2
in Eq. (6-33). The sin-
gle-chain structure functionsS
A(k) and
S
B(k) are expressed for Gaussian chains by
the Debye functionf
D(x)as
S
A(k)=N
Af
D(k
2
·R
2
gyr,A
Ò)
S
B(k)=N
Bf
D(k
2
·R
2
gyr,B
Ò) (6-38)
wheref
D(x)∫2[1–(1–e
–x
)/x]/x.
Eqs. (6-36) and (6-37) result (Binder,
1983) from random phase approximations
(De Gennes, 1979) and agree with the
treatment as given by Eqs. (6-9)–(6-30)
(but using Eq. (6-32) instead of Eqs. (6-4)–
(6-6)) only in the long wavelength limit,
wherek
c·R
2
gyr
Ò
1/2
<1 (for both types of
chains). In this limit
L
kcan be replaced
by a constant
L
0, and the results are just
special cases of Eqs. (6-9)–(6-30); e.g., for
a symmetric mixture (N
A=N
B,s
A=s
B=a)
we simply find:
Hence the critical wavelength
l
cis rescaled
by a large prefactor, namely the coil gyra-
tion radius, similar to the correlation length
x
coexof concentration fluctuations at the
coexistence curve,
x
coex=(Na
2
/36)
1/2
(c/c
c–1)
–1/2
(6-40)
Eq. (6-39), however, is only applicable for
“shallow” quenches, where
cdoes not
greatly exceed
c
s(F). A different behavior
occurs (Binder, 1983) for deep quenches,
where
coc
s(F) (thus describing the un-
mixing of two incompatible polymers):
(6-41)
Now the initial wavelength of maximum
growth is smaller than the coil radius,
andk
mOk
c, since the gradient-square ex-
pansion, Eq. (6-32), is no longer appli-
cable.
As is obvious from Eq. (6-36), the relax-
ation rateR(k) is the product of two fac-
tors, a static factor [S
T
coll(k)]
–1
which con-
tains the thermodynamic singularities at
the critical point (and along the spinodal
curve; cf Eq. (6-39)), and a kinetic factor
k
2
L
k, to which we now turn. Here the fac-
tork
2
simply reflects the conservation law
for the concentration, while
L
kreflects the
k
Na
k
Na
cs
ms≈
⎛
⎝
⎜
⎞
⎠
⎟
≈
⎛
⎝
⎜
⎞
⎠
⎟
6
2
6
2
2
12
12
2
12
14
/
/
/
/
[/()]
[/()]cc
cc
Φ
Φ
2
18
1
2
2
12
12
p/(()/),
/
/
/
lc c
cc s
mc==
=( 6-39
)
k
Na
kk
⎛
⎝
⎜
⎞
⎠
⎟ − Fwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

479. complicated dynamics of single polymer
chains in an entangled polymer melt. De-
spite some effort (De Gennes, 1980; Pin-
cus, 1981; Binder, 1983; Akcasu et al.,
1986) there is no systematic method that
yields information on thek-dependence of
L
k; the approximate theories quoted above
all lead to a long-wavelength limit called
theslow-mode theory, which is believed to
be an incorrect description of interdiffusion
because it disregards bulk flow (Kramer
et al., 1984). Remember that Eq. (6-36) for
kÆ0 can be interpreted asR(k)=k
2
D
coll
,
whereD
coll
=L
0/S
T
coll(0), i.e., uphill diffu-
sion. In the slow-mode theory, the interdif-
fusion coefficient of an ideal non-interact-
ing mixture is expressed in terms of tracer
diffusion coefficientsD
t
AandD
t
Bof an A
chain (B chain) in a pure B (A) environ-
ment as
(D
coll
)
–1
=(1–F)/D
t
A+F/D
t
B (6-42)
i.e., the slow species controls the interdif-
fusion coefficients. In contrast, thefast-
mode theory(Kramer et al., 1984) yields
the opposite result, that the fast species
controls interdiffusion:
D
coll
=(1–F)D
t
A+FD
t
B (6-43)
Experimental evidence (for a review of
polymer interdiffusion, see Binder and Sil-
lescu, 1989) seems to favor Eq. (6-43) over
Eq. (6-42), but we feel that a valid deriva-
tion of Eq. (6-43) is still lacking, and
moreover neither Eq. (6-43) nor (6-42) de-
scribe interdiffusion in lattice gas models
correctly, as computer simulations show
(Kehr et al., 1989). Both Eqs. (6-42) and
(6-43) are based on a description of trans-
port in which in the constitutive equation
relating currents and chemical potential
differences (cf. Eq. (6-11))off-diagnonal
Onsager coefficients are neglected; the
Monte Carlo simulation shows that this ap-
proximation is inaccurate.
IfD
t
AandD
t
Bare of the same order, then
Eqs. (6-42) and (6-43) also yieldD
coll
of
the same order, and the order of magnitude
ofD
coll
(and henceR(k), Eq. (6-36)) is then
predicted correctly from these equations. If
the chain lengthN
A(N
B) is less than the
chain lengthN
e
A
(N
e
B
) between effective
entanglements, the single-chain dynamics
are simply given by the Rouse (1953)
model, which implies
D
t
A≈s
AW
A/N
A,D
t
B≈s
BW
B/N
B
whereW
A(W
A) are time constants for the
reorientation of subunits. In contrast, for
N
ApN
e
A
,N
BpN
e
B
the reptation model
(Doi and Edwards, 1986; De Gennes,
1979) implies for strongly entangled chains
D
t
A≈s
AW
AN
e
A
/N
2
A
,D
t
B≈s
BW
BN
e
B
/N
2
B
(neglecting prefactors of order unity). A
problem which is not completely under-
stood so far is the concentration depen-
dence of the parametersN
e
A
andN
e
B
in
blends. In view of all these uncertainties
about the validity of Eqs. (6-42) and (6-43)
and the precise values of the constantsD
t
A
andD
t
Bto be used in them, it is better to
consider
L
kin Eq. (6-36) as a phenomeno-
logical coefficient, about which only the
order of magnitude can roughly be in-
ferred.
An effect disregarded by the theories but
seen in computer simulations (Sariban and
Binder, 1989a) is the change in the chain
linear dimensions (Eq. (6-31)) in quench-
ing experiments. This also means that the
parameterain Eqs. (6-32) and (6-39) to
(6-41) should be treated as a
c-dependent
quantity, which further complicates the
quantitative analysis of experiments.
6.2.5 Significance of the Spinodal Curve
From Eq. (6-25) it is evident that the
linear theory predicts a singular behavior
428 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

480. 6.2 General Concepts 429
when the concentrationc
–
approaches the
concentrationc
sp(T) of the spinodal curve,
defined by
(∂
2
f
cg/∂c
2
)
T,c
–
=c
sp(T)= 0 (6-44)
Here the critical wavelength
l
cdiverges to
infinity. For example, if we adopt Eq. (6-8)
withA~(T/T
c
MF–1), near the spinodal we
obtain, omitting prefactors of order unity,
On the metastable side of the spinodal
curve a similar divergence occurs for both
the correlation length
x(c
–
) of concentra-
tion fluctuations and the radiusR* of a crit-
ical droplet, namely (Binder, 1984b)
AlthoughR* diverges asc
–
approaches
c
sp(T), the associated free-energy barrier
DF* against the formation of a critical
droplet vanished there (Fig. 6-9). The max-
imum growth rate of spinodal decomposi-
x()~ *~ ( / )
()
,()
/
/
cRr TT
cT c
cc
ccT
1
12
12
−
×
−
−
⎡
⎣
⎢
⎤
⎦
⎥
−
−
c
MF
sp
coex
(2)
coex
(1) sp
(6-46)
Á
l
cc
MF sp
crit sp
sp
(6-45)
~( / )
()
()
()
/
/
rTT
ccT
ccT
ccT
1
12
12
−
−
−
⎡
⎣
⎢
⎤
⎦
⎥
−
−
Ê
tion, which according to Eqs. (6-21), (6-25)
and (6-26) can be written as
R
m= (1/4)Mr
2
k
BT(2p/l
c)
4
(6-47)
vanishes asc
–
Æc
sp(T), and a similar “criti-
cal slowing down” (Hohenberg and Halpe-
rin, 1977) would occur in the growth rate of
a supercritical droplet. Thus, within the
framework of this theory, the spinodal
curve plays the role of a line of critical
points. We now wish to investigate whether
there is a physical signficance to this singu-
lar behavior.
In Sec. 6.2.1 we emphasized that the def-
inition off
cgis not unique but really in-
volves a length scaleLover which a coarse-
graining of short-range fluctuations is per-
formed. This is best seen from the attempts
to calculatef
cgexplicitly, which can be
approximated by using renormalization
group methods (Kawasaki et al., 1981) or
by Monte Carlo simulation (Kaski et al.,
1984). These treatments show that the posi-
tion ofc
sp(T) depends strongly on the
length scaleL(see Fig. 6-10a), and there-
fore for systems with short-range forces, to
which these treatments apply, there is no
physical significance to the spinodal singu-
larity (Eqs. (6-44) to (6-47)) whatsoever.
Figure 6-9.(a) Character-
istic lengthsR*,
x,l
cand
(b) nucleation barrierDF*
vs. concentrationc
B(sche-
matic). Full curves are the
predictions of the linearized
Cahn–Hilliard mean-field
theory of nucleation. Dash-
dotted curves show,on a
different scale, the conjec-
tured smooth behavior of a
system with extremely
short-range interaction, for
which the spinodal singu-
larity is completely washed
out (Binder, 1981).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

481. 430 6 Spinodal Decomposition
Figure 6-10.(a) Monte Carlo results for the dependence of the relative distance of the “spinodal” from the co-
existence curve on the size of the coarse-graining cellL. The results refer to a simple-cubic nearest-neighbor
Ising magnet in the critical region, and are obtained from sampling the magnetization distribution function
P
L(M)inL¥L¥Lsub-blocks of a 24
3
system. Here,S
maxis the value at which lnP
L(M) has its maximum, and
corresponds to the coexistence curve if we assume lnP
L(M)~L
3
f
cg/k
BT,withc =(1–M )/2, and the “spinodal”
is estimated as inflection pointM
sof lnP
L(M). By scalingLwith the correlation length x, all temperatures
superimpose on one “scaling function” (Kaski et al., 1984). (b) Extrapolation of the inverse collective structure
function versus inverse temperature to locate the spinodal temperaturesT
sp(c
–
) or their inverse (arrows). Here
Monte Carlo simulation data for a polymer mixture are used, the polymers (A, B) being modelled as self- and
mutually avoiding random walks on the simple cubic lattice withN
A=N
B=N= 32 steps, at a concentration of
vacancies
f
v= 0.6. Values on the curves are the reduced volume fractionF
A/(1–F
v) of monomers of A chains.
If two neighboring lattice sites are taken by monomers of the same kind, an energy
eis obtained, and thus an en-
thalpic driving force for phase separation is created. From Sariban and Binder (1989b). (c) Phase diagram of the
model for a polymer mixture as described in (b), displaying both the true coexistence curve (binodal) and the
extrapolated spinodal. From Sariban and Binder (1989b).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

482. 6.2 General Concepts 431
Another way of showing this comes from
a closer examination of the procedures
by which experimentalists locate spinodal
curves. Eqs. (6-30) and (6-37) imply that
[S
T
coll(kÆ0)]
–1
~(∂
2
f
cg/∂c
2
)
T,c
–~T–T
sp(c
–
)
whereT
sp(c
–
) is the inverse function of
c
sp(T)inthec–Tplane. Thus a plot of
[S
T
coll(kÆ0)] vs. temperature should allow
a linear extrapolation to locateT
sp(c
–
) from
the vanishing of [S
T
coll(kÆ0)] (cf. Fig. 6-
10b). However, the extrapolated spinodal
determined in this way (e.g., Sariban and
Binder, 1989b) (Fig. 6-10c), is not physi-
cally meaningful as it crosses the true co-
existence curve (binodal) nearT
c, which is
physically impossible. This happens be-
cause, in reality,
[S
T
coll(kÆ0)]
–1
c
–
=c
crit
~(T–T
c)
g
withg≈1.24, a critical exponent different
from the mean-field result
g= 1. Hence
a linear extrapolation fails nearc
–
=c
crit.At
strongly off-critical concentrations this ex-
trapolation is also ambiguous, because usu-
ally actual data cannot be taken deep in the
metastable phase for temperatures close to
T
sp(c
–
).
This situations is different, however, for
systems with infinitely weak but infinitely
long-range interactions (Penrose and Lebo-
witz, 1971); then the mean-field theory is
correct because statistical fluctuations are
suppressed. At the same time, the lifetime
of metastable states is infinite because ho-
mogeneous nucleation is no longer pos-
sible (“heterophase fluctuations” are su-
pressed at the same times as “homophase
fluctuations”). The spinodal curve is the
limit of metastability here.
While such a system with infinitely
long-range interactions is clearly artificial,
it makes sense to consider systems with
long but finite rangerof the interactions
(Heermann et al., 1982; Binder, 1984b;
Heermann, 1984a,b). Although very close
to the critical pointT
csuch system behave
qualitatively like short-range systems (this
is to be expected from the so-called “uni-
versality principle” (Kadanoff, 1976)), far-
ther fromT
ca well-defined mean-field crit-
ical region exists. It is this region where
both the coarse-graining defined in Eq. (6-
1) and the linearization approximation of
Sec. 6.2.2 make sense, as we will now dis-
cuss.
The argument is simply an extension of
the Ginzburg (1960) criterion for the valid-
ity of the mean-field theory for critical
phenomena to nucleation and spinodal de-
composition (Binder, 1984b): nonlinear
terms indc(x,t) can be neglected if their
mean-square amplitude in a coarse-grain-
ing cell is small in comparison with the
concentration difference squares in the
system, over which relevant nonlinear ef-
fects are felt:
·[dc(x,t)]
2
Ò
T,L[c
–
–c
sp(T)]
2
(6-48)
We estimate·[dc(x,t)]
2
Ò
T,Las
·[dc(x,t)]
2
Ò
T,L≈·[dc(x,0)]
2
Ò
T,Lexp[2R
mt]
whereR
mis the maximum growth rate de-
fined in Eq. (6-26), and the initial mean-
square amplitude is related to the correla-
tion function of concentration fluctuations
in the initial state at temperatureT
0. Thus,
using Eq. (6-1):
(6-49)
the summations overiandjbeing restrict-
ed to sites within the cell of sizeL
3
cen-
tered atxand in the last step the sums are
converted into integrals (one sum
Â
i
is can-
celled against a factorL
3
making use of the
〈〉〈〉
〈〉−
〈〉−
∑
∫
[(,)] [()]
[]
[()() ]
,,
,ddcc
L
cc c
L
cc c
TL T L
ij
ijT
Txx
xx
0
1
1
0
22
6
2
3
2
0
0
0
=
=
=dwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

483. translational invariance of the correlation
function). For distances|x|
xthe correla-
tion function is simply given by a power-
law decay,·c(0)c(x)Ò
T
0
–c
–2
≈r
–2
x
–1
,and
therefore the order of magnitude of the in-
tegral in Eq. (6-49) is estimated, yielding
·[dc(x, 0)]
2
Ò
T
0
,L≈r
–2
L
–1
(6-50)
As expected, the mean-square concentra-
tion fluctuation is the smaller the larger is
the range of the interactions and the larger
is the coarse-graining lengthL. Since the
largest permissible value forLisL≈
l
c,
Eqs. (6-45), (6-48), and (6-50) readily
yield
exp (2R
mt)Or
3
(1 –T/T
c)
1/2
¥[c
–
/c
sp(T)–1]
3/2
(6-51)
A similar self-consistency criterion can be
formulated in the metastable region, where
the largest permissible choice forLin the
Ginzburg criterion
·[dc(x)]
2
Ò
T,LO(c
sp(T)–c
–
)
2
now isL= x(c
–
), as given in Eq. (6-46), and
it is found in full analogy with Eqs. (6-48)
to (6-51) that
1Or
3
(1 –T/T
c)
1/2
[1 –c
–
/c
sp(T)]
3/2
(6-52)
It is also interesting that Eq. (6-52) can be
derived from a completely different argu-
ment, namely requiring that the free energy
barrier of nucleationD
∫*pk
BT(Binder,
1984b) (forc
–
nearc
sp(T) the Cahn–Hil-
liard (1959) mean-field theory of nuclea-
tion predictsD
∫*/k
BT~r
3
(1 –T/T
c)
1/2
¥[1 –c
–
/c
sp(T)]
3/2
). From Eqs. (6-51) and
(6-52) we draw the following conclusions:
(i) For the linearized theory of spinodal
decomposition to be self-consistent, and for
the description of a metastable state near
the spinodal to be self-consistent (and that it
has a long lifetime owing to a large barrier
against homogeneous nucleation), the in-
equalities which must be satisfied require
that
r
3
(1 –T/T
c)
1/2
p1 (6-53)
This condition can be satisfied only for
a large rangerof interaction, and defines
themean-field critical region(Ginzburg,
1960). As long as Eq. (6-53) holds, the
Landau mean-field theory of critical phe-
nomena (Stanley, 1971) is quite adequate.
However, for systems with rather short-
range interactions Eq. (6-53) and hence
Eqs. (6-51) and (6-52) can never be ful-
filled.
(ii) Even for systems with a long but fi-
nite rangerof interaction, close enough to
T
c(namely forr
3
(1 –T/T
c)
1/2
≈1) a cross-
over occurs to a non-mean-field critical be-
havior, described by the same critical be-
havior as short-range systems. This is ex-
pected from the principle of universality of
critical behavior (Kadanoff, 1976). Then
the nonlinear character of Eq. (6-15) is im-
portant during the initial stages of the
quench, and the linearization approxima-
tion (Eq. (6-18)) is never warranted.
(iii) For mean-field-like systems for
which Eq. (6-53) holds, Eqs. (6-51) and
(6-52) hold only if we do not come too
close to the spinodal. The region excluded
by these inequalities scales as
|c
–
/c
sp(T)–1|~r
–1
(1 –T/T
c)
–1/3
In this excluded region the singularities de-
scribed by Eqs. (6-45) to (6-47) are essen-
tially smeared out, and a gradual transition
from nucleation and growth to nonlinear
spinodal decomposition occurs. This grad-
ual transition can be understood qualita-
tively from a cluster dynamics treatment of
phase separation (Binder and Stauffer,
1976a; Binder, 1977, 1981; Mirold and
Binder, 1977; Binder et al., 1978). Figs.
6-11 and 6-12 summarize the main ideas
about this description which is closely
432 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

484. 6.2 General Concepts 433
related in spirit to the Langer–Schwartz
(1980) and Kampmann–Wagner (1984)
treatments of concomitant nucleation,
growth, and coarsening (see Wagner et al.,
2001). Each state of the system is charac-
terized by a cluster size distributionn
–
l(t),
wherelis the number of B atoms contained
within a cluster (Fig. 6-11a) and the bar
represents an average over other “cluster
coordinates” (cluster surface area, shape,
etc.) which are not considered explicitly.
The time evolution of the cluster size
distribution in a quenching experiment
is described by a system of kinetic equa-
tions:
HereS
l+l¢,l¢ is a rate factor for a splitting
reaction of a cluster of sizel+l¢into two
d
d
=
(6
=
=
=
=t
nt S n t
Snt
cntnt
cntnt
l
l
lll ll
l
l
lll
l
l
lll l ll
l
lll l() ()
()
() ()
() ()
,
,
,
,
′
∞
+′′+′
′
−
′
′
−
−′′ ′ −′
′
∞
′′∑
∑
∑
∑
−
+
−
1
1
1
1
1
1
1
2
1
2
--54)
clusters of sizelandl¢, andc
l,l¢is the rate
factor for the inverse “coagulation” reac-
tion. These rate factors are assumed to be
independent of timetand hence Eq. (6-54)
also describes the concentration fluctua-
tions in thermal equilibrium, where de-
tailed balance must hold between splitting
and coagulation reactions:
S
l+l¢,l¢n
l+l¢=c
l,l¢n
ln
l¢∫W(l,l¢) (6-55)
wheren
lis the cluster concentration in
equilibrium andW(l,l¢) is a cluster reac-
tion matrix in equilibrium. Whilen
l(t=0)
for a random distribution of atoms in the
alloy is just the cluster distribution in the
well-known percolation problem (Stauffer,
1985),n
landW(l,l¢) are not explicitly
known, but can be fixed by plausible as-
sumptions (Binder, 1977; Mirold and
Binder, 1977). Then Eqs. (6-54) and (6-55)
can be solved numerically (see Fig. 6-11b).
What must happen is thatn
–
l(t) fort Æ∞
develops towardsn
l, the cluster size distri-
bution in the state at the coexistence curve
c
(1)
coex
,withc
(1)
coex
= ln
l. The excess
concentrationc
–
–c
(1)
coex
=l[n
l(0) –n
l]is
l=1
∞
∑
l=1
∞
∑
Figure 6-11.(a) “Clusters”
of B atoms in a binary AB
mixture defined from con-
tours around groups of B
atoms and their reactions.
(b) Cluster size distribution
n
–
l(t) for various timestafter
a quench from infinite tem-
perature toT/T
c=0.6 at
c
–
= 0.1 for parameters appro-
priate for a two-dimensional
Ising model of a binary al-
loy. These results were ob-
tained from a numerical so-
lution of Eq. (6-54), where
time units are rescaled by an
arbitrary rate factor. From
Mirold and Binder (1977).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

485. redistributed into macroscopic B-rich do-
mains (of concentrationc
(2)
coex
), i.e., occurs
at cluster sizel
Æ∞fortÆ∞. For interme-
diate times, we find a nonmonotonic clus-
ter size distribution (Fig. 6-11b); a mini-
mum occurs at thecritical size l* of nucle-
ation theory (Binder and Stauffer, 1976a),
while the broad maximum at larger sizes is
due to growing supercritical clusters which
have already been nucleated. Forl<l* the
cluster size distribution is basically the
equilibrium size distribution of a slightly
supersaturated solid solution. As time goes
on, the peak inn
–
l(t), representing the
supercritical growing clusters, shifts to
larger and larger cluster sizes, and at the
same time the supersaturation is dimin-
ished, until fort
Æ∞the peak has shifted
tol
Æ∞. This separation of clusters into
two classes – small ones describing con-
centration fluctuations in the supersatu-
rated A-rich background and large ones
describing the growing B-rich domains –
can also be shown analytically from Eqs.
(6-54) and (6-55) (Binder, 1977). This pic-
ture of the phase separation process also
emerges very clearly from computer simu-
lations (Marro et al., 1975; Rao et al.,
1976; Sur et al., 1977).
The description in terms of Eqs. (6-54)
and (6-55) contains both nucleation and
growth and coagulation as special cases
(Binder, 1977), and it can be used as a ba-
sis for understanding thegradual transi-
tion from nucleation to spinodal decompo-
sition(Binder et al., 1978). In the meta-
stable regime the density of unstable fluc-
tuations (“critical” and “supercritical” clus-
ters withl<l*) is very small (Fig. 6-12a),
because the energy barrierDF*pk
BT.
Near the spinodal curve, on the other hand,
DF*≈k
BT, and hence there is a high den-
sity of unstable fluctuations:DF*hereno
longer limits the growth, but rather the
conservation of concentration. Near grow-
ing clusters,c(x,t) locally decreases, and
then no other cluster can grow there. Ow-
ing to this “excluded volume” interaction
of clusters, a quasiperiodic variation of
concentration results (Fig. 6-12a, bottom),
roughly equivalent to a wavepacket of
Cahn’s concentration waves. But the latter
are not growing independently, rather they
are strongly interacting; hence it is more
434 6 Spinodal Decomposition
Figure 6-12.(a) Schematic “snapshot
pictures” of fluctuations of an unmixing
system in the metastable regime (top)
and in the unstable regime (bottom).
Unstable fluctuations, which steadily
grow with increasing time (arrows) are
shaded. (b) Structure factorS(q,t)vs.
scaled wavevectorqat various timest
after a quench from infinite temperature
toT=0.6T
cforc
–
= 0.1. Parameters are
chosen for a three-dimensional Ising
model of a binary alloy. These results
were obtained from a numerical solu-
tion of Eqs. (6-54) to (6-56). From
Binder et al. (1978).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

486. 6.2 General Concepts 435
reasonable to consider large clusters as
objects nearly independent of each other
(apart from the reactions in Eq. (6-54))
rather than as these waves.
The fact that the cluster picture and the
concentration picture are simply two dual
descriptions of the same physical situation
becomes very apparent when the structure
factorS(k,t) is calculated from the cluster
size distributionn
–
l(t). Introducing the con-
ditional probabilityg
l(x) that the sitex+x¢
is taken byaBatomif x¢is taken by a B
atom of anl-cluster, we find (Binder et al.,
1978):
(6-56)
Simple assumptions forg
l(x) yieldS(k,t)
as shown in Fig. (6-12b). The resulting
S(k,t) is therefore in qualitative agreement
with experimentation (Fig. 6-5a, b), com-
puter simulation of microscopic models
(Fig. 6-5c,d) and the approximate non-
linear theory of spinodal decomposition
of Langer et al. (1975) (see Fig. 6-8, bot-
tom).
This gradual transition between nuclea-
tion and spinodal decomposition always
appears close to the critical point in the
phase diagram (Fig. 6-13); for systems
with a large interaction rangera mean-
field critical regime exists (Eq. (6-53))
where this gradual transition is confined to
a narrow regime adjacent to the spinodal.
Outside this transition regime the linear-
ized theory of spinodal decomposition is
expected to holdinitiallyin the unstable re-
gime, while a nonclassical nucleation the-
ory (where ramified droplets, which must
first be compacted before they can grow,
are nucleated (Klein and Unger, 1983;
Unger and Klein, 1984; Heermann and
Klein, 1983a, b)) holds in the metastable
regime. This crossover line between clas-
Skt ln t c
l
ll
(,) () { () }
exp ( )
=d
i
=1
∞
∑ ∫ −
×⋅
xx
kx
g
sical and nonclassical “spinodal nuclea- tion” is essentially given by the condi- tion that the free-energy barrierDf*to
form a critical nucleus is of the order of r
3
(1 –T/T
c)
1/2
p1 (Fig. 6-14a). Therefore
this regime of “spinodal nucleation” must disappear when the non-mean-field critical region is approached (Fig. 6-14b).
It should be emphasized that the phase
diagram in Fig. 6-13 showing these various regimes in the temperature–concentration plane is relevant only in the very early stages of phase separation. This should be obvious from Eq. (6-51). The time range where it is valid is, at best, of the order of R
mt≈lnr(rbeing measured in units of the
lattice spacing). Therefore the experimental verification of the linear theory of spinodal decomposition is a difficult problem (see Sec. 6.4). Clear confirmation of the ideas described above has come from computer simulations of medium-range Ising models due to Heermann (1984a) (see Figs. 6-15 and 6-16). Since for the Ising model the re- sult forR(k) is readily worked out, a com-
parison between theory and simulation is possible without any adjustable parameter whatsoever! Fig. 6-15 shows that there is indeed a regime where initially fluctuations increase exponentially with time (Eq. (6-23)) and the observed ratesR(k)do
agree with the predictions, if we stay far off the spinodal.
It is not easy to identify physical systems
that have a large but finite interaction range. What is really needed is a large pre- factor in the mean-field result for the corre- lation length (Eq. (6-46)), and such a large prefactor has in fact been identified for polymers with high molecular weight (Eq. (6-40)). This happens because for polymers the coefficient of the gradient energy in Eq. (6-32) comes from the random-coil struc- ture, and does not have the meaning of a squared interaction range as in Eq. (6-4).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

487. 436 6 Spinodal Decomposition
Figure 6-13.Various regimes in the phase diagram of a binary mixture AB, showing part of the plane formed
by temperatureTand volume fraction
Fof the B component (only volume fractionsF<F
critare shown, since
this schematic phase diagram is symmetric around the axis
F=F
crit). The horizontal broken line separates the
non-mean-field critical regime (top) from the mean-field critical regime (bottom). The two solid curves are the
coexistence curve (left) and the spinodal curve (right). The two dash-dotted curves on both sides of the spino-
dal limit represent the regime where a gradual transition from nucleation to spinodal decomposition occurs. The
linearized theory of spinodal decomposition (Fig. 6-1a) should hold to the right of these dash-dotted curves. The
regime between the coexistence curve and the left of the two broken curves is described by classical nucleation
theory (compact droplets, Fig. 6-1b). The regime between the right broken curve and the left dash-dotted curve
is described by “spinodal nucleation” (ramified droplets). From Binder (1984b).
Figure 6-14.Schematic plots of the nucleation free energy barrier for (a) the mean-field critical region
of ad-dimensional alloy system, i.e.,r
d
(1 –T/T
c)
(4 –d)/2
o1 and (b) the non-mean-field critical region,
i.e.,r
d
(1 –T/T
c)
(4 –d)/2
< 1. The gradual transition from nucleation to spinodal decomposition occurs for
DF*/k
BT
c≈1. From Binder (1984b).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

488. 6.2 General Concepts 437
However, we note that forN
A=N
B=Nwe
may map Eqs. (6-33) and (6-32) into Eqs.
(6-4) and (6-6) with the following identifi-
cations (Binder, 1984b):
(6-57)
The condition that the nucleation barrier
D
∫*/k
BTp1 instead of Eq. (6-52) then
yields
(6-58)
and a similar criterion holds on the un-
stable side of the spinodal (Binder, 1983),
with 1 being replaced by exp(2R
mt)on
the left-hand side of Eq. (6-58), as in Eq.
(6-51). It follows that for large chain length
Nthe spinodal is smeared out over a region
of widthN
–1/3
.
In view of these results, it is gratifying to
note that for polymers, convincing experi-
mental demonstrations of the validity of
Cahn’s linearized theory of spinodal de-
composition have indeed been presented
(see Sec. 6.3.4).
6.2.6 Towards a Nonlinear Theory
of Spinodal Decomposition
in Solids and Fluids
In the last subsection it was shown that
the regime of times for which the linear
theory of spinodal decomposition holds is
extremely restricted, if it holds at all.
Therefore, a treatment of nonlinear effects
is necessary. A systematic approach is only
possible by an expansion in powers of 1/r
for larger(Grant et al., 1985). This theory
is very complicated and has so far been
worked out only forc
–
=c
crit. It explicitly
shows the coupling between concentration
waves with different wavevectors.
11 1
12
12
32
ONccT
/
/
/
/()−
⎛
⎝
⎜
⎞
⎠
⎟−
c
c
crit
sp
||
DD∫∫
polymer small molecules
c ≡
≡≡
1
2
12
3
22
N
T
TN
rNa
c ,
Figure 6-15.Time dependence of the logarithm of
spherically averaged structure factors of a 60
3
sim-
ple-cubic Ising lattice, where each spin interacts with
q= 124 neighbors with equal interaction strengthJ,
quenched from infinite temperature to
T=(4/9)T
c
MF=(4/9)qJ/k
Bandc
–
= 0.4
The five smallest wavenumbersk
n=2pn/60 are dis-
played. Straight lines for short times indicate expo-
nential growth and thus yieldR(k). From Heermann
(1984a).
Figure 6-16.Cahn plotR(k)/k
2
vs.k
2
(R(k)isde-
noted as
w(k) in this figure), as extracted from data
such as shown in Fig. 6-15. Crosses are the Monte Carlo results, and straight lines are the predictions of the linearized theory. Note thatc
sp(T)≈0.127 in this
case. From Heermann (1984b).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

489. So far the most popular (if approximate)
approach is the decoupling approximation
suggested by Langer et al. (1975). Exact
equations of motion are derived for the
probability distributions
r
1[c(x)],r
2[c(x
1),
c(x
2)], etc. Herer
1is the probability den-
sity that, at pointx, the concentrationc(x)
occurs, and
r
2is the corresponding two-
point function. As expected, the equation
of motion for
r
1involvesr
2, and the equa-
tion of motion for
r
2involves the three-
point function
r
3, etc., so that an infinite
hierarchy of equations of motion is gener-
ated.
This hierarchy is decoupled by the fol-
lowing approximation for the two-point
function [dc(x)=c(x)–c
–
]:
(6-59)
The motivation for Eq. (6-59) is the fol-
lowing: if there were no correlation be-
tween concentrations at pointsx
1andx
2,
the probability
r
2would just be the product
of the one-point probabilities. Therefore,
the correction of this factorization approxi-
mation is made proportional to the two-
point correlation function·dc(x
1)dc(x
2)Ò
T.
In this way, Eq. (6-59) yields a closed equa-
tion of motion for the probability
r
1. This
equation is then solved approximately, as-
suming that the coarse-grained free energy
has the Landau form (Eqs. (6-5) and (6-8)).
The coefficientsA,Bandrare adjusted
self-consistently such that the resulting
state equilibrium is correctly described in
the non-mean-field critical region.
This approach has also been worked out
for the dynamics of non-conserved order
parameters (Billotet and Binder, 1979) and
the validity of this approximation has been
studied carefully (see also Binder et al.,
1978). The final result can be cast in a form
rrr
21 2 1112
12
2
12
2
1
[(),()] [()][()]
() ()
()
() ()
()
cc c c
cc
c
cc
c
T
TT
xx x x
xx xx
=
×+
〈〉
〈〉 〈〉
⎧
⎨
⎩
⎫
⎬
⎭
dd
d
dd
d
that is similar to Eq. (6-27), namely,
where all nonlinear effects are now con-
tained in a correction terma(t), which it-
self depends onS(k,t) in a nonlinear way.
As noted above, the coefficientsA,B,
andrin Eqs. (6-5) and (6-8) are adjusted
such that the critical behaviors of the co-
existence curve, critical scattering inten-
sity, and correlation length (at the coexis-
tence curve) are reproduced:
(c
(1)
coex
–c
crit)/c
crit=B
ˆ
(1 –T/T
c)
b
(6-61a)
c
coex=C
ˆ
(1 –T/T
c)
–g
(6-61b)
x
coex=x
ˆ
(1 –T/T
c)
–n
(6-61c)
whereB
ˆ
,C
ˆ
, and
x
ˆ
are the appropriatecriti-
cal amplitudesand
b,g, andnthe asso-
ciatedcritical exponents(Stanley, 1971;
Binder, 2001). It now turns out that the
strength of nonlinear effects is controlled
by the inverse of a parameterf
0which near
T
cis expressed as (Billotet and Binder,
1979)
f
0~x
ˆd
B
ˆ
2
C
ˆ
–1
(1 –T/T
c)
g+2b–dn
(6-62)
for ad-dimensional system. In the non-
mean-field critical region, the hyperscaling
relationd
n=g+2b(Kadanoff, 1976) elimi-
nates the temperature dependence from Eq.
(6-62). In addition, two-scale factor uni-
versality (Stauffer et al., 1972) implies that
the critical amplitude ratio
x
ˆd
B
ˆ
2
/C
ˆ
and
hencef
0is a universal constant of order
unity (Billotet and Binder, 1979), i.e.,
f
0≈9.45. On the other hand, if we consider
a quench into the mean-field critical re-
gion, again adjusting the coefficientsAand
Bin Eq. (6-8) but now using mean-field
d
d
= (6-60)
B
t
Skt Mk
f
c
at rTk Skt kT
Tc
(,)
() (, )
,
−
×
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟++
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
−
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
2
2
2
2
22
438 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

490. 6.2 General Concepts 439
results in Eqs. (6-61a–c), namely,b=
1
–
2
,
g=1, andn=
1
–
2
,x
ˆ
~r, we obtain
f
0~r
d
(1 –T/T
c)
(4–d)/2
forr
d
(1 –T/T
c)
(4 –d)/2
p1 (6-63)
It is seen thatf
0is now simply proportional
to the parameter appearing in the Ginzburg
criterion, and hence very large. For
r
d
(1 –T/T
c)
(4 –d)/2
≈1 a crossover to the uni-
versal constantf
0≈9.45 in the non-mean-
field critical region occurs.
Fig. 6-17 shows the time evolution for
the structure factor for two choices off
0.
Here again a rescaling of the structure fac-
torsS
ˆ
=r
2
k
c
2Sand of timet=2Mr
2
Tk
c
4t
was used, whileq=40k/(2pk
c) here. It is
seen that forf
0≈9.45 the linear theory is
indeed invalid from the start, as expected.
However, for large values off
0the linear
theory does hold initially, consistent with
the simulations (Fig. 6-15). The same con-
clusion emerges from the time dependence
of the rescaled effective second derivative
m
–(T,t) in Eq. (6-60):
m
–(T,t)∫1+a(t)/[∂
2
f/∂c
2
]
T,c (6-64)
(see Fig. 6-18). Nonlinear effects are negli-
gible as long as
m
–= 1, while the decrease
in
m
–is a signal of nonlinear effects, since
a(t) is always negative. The physical
significance of
m
–(T,t) is that it describes
the ratio between the actualk
c
2(t), where
the growth rate in Eq. (6-60) changes sign,
and the corresponding prediction of the
Cahn–Hilliard theory. Fig. 6-18 implies
thateven if the nonlinear effects are very
strongthe Cahn–Hilliard prediction fork
c
2
differs from the actualk
c
2(t) by about a fac-
tor of two at most; therefore, the Cahn–
Hilliard theory is certainly useful for esti-
Figure 6-17.Plot of ln {[S
˜
(q, t)–S
˜
T(q)]/S
˜
T
0
(q)} vs.
scaled time
tfor five different values ofq,fora
quench from infinite temperature atc
crit, using
f
0= 9.45 (top) andf
0= 9450 (bottom). From Carmesin
et al. (1986).
Figure 6-18.Rescaled effective second derivative
m
–(T
f,t) (Eq. (6-64)) vs. scaled timetfor four
choices off
0. Only whenm
–(T,t)≈1 is the Cahn–
Hilliard–Cook approximation accurate. Forf
0= 9.45
the initial value
m
–(T, 0) = 0.65. From Carmesin et al.
(1986).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

491. mating the order of magnitude of the range
of wavelengths that is unstable initially.
For systems with short-range forces,
wheref
0≈9.45 applies, the early-time
structure factor for critical quenches (Fig.
6-8, bottom) is in reasonable agreement
with both experiment (Fig. 6-5a, b) and
simulations (Figs. 6-5c, d). However, the
Langer–Baron–Miller (LBM) (1975) ap-
proximation gets worse for strongly off-
critical quenches (Binder et al., 1978). The
theory again exhibits a spurious singularity
at a spinodal curve, which does not coin-
cide with the mean-field spinodal of the
CHC approximation, but occurs at a renor-
malized concentration closer to the coexis-
tence curve. This shift of the spinodal is not
surprising, since the effective second de-
rivative of the potential is also renormal-
ized (Fig. 6-18). This spinodal of the LBM
approximation is completely spurious – its
precise location depends on details of the
coarse-graining procedure. For concentra-
tions between the coexistence curve and
this spinodal, i.e., the metastable regime,
the structure factorS(k,t) forTÆ∞satu-
rates at a valueS(k,∞)=k
BT/[∂
2
f/∂x
2
)
T,c
+a(∞)+r
2
Tk
2
], i.e., at the Ornstein–Zer-
nike result for scattering from fluctuations
in metastable equilibrium (Binder et al.,
1978).
We now consider fluid binary mixtures.
Here the theoretical formulation is com-
plicated by another long-range interaction,
namely the hydrodynamic backflow inter-
action. If the Liouville equation is written
for the probability distribution
r({c(x)},t)
that a concentration fieldc(x) occurs,
(6-65a)
the Liouville operator
∞contains a term
∞
Aalso present in solid alloys and another
term
∞
HDcontaining the Oseen tensor
T={T
ab} describing the hydrodynamic
∂
∂t
ct ct
rr({ ( )}, ) ({ ( )}, )xx= ∞
interaction:
and
where
his the shear viscosity. Kawasaki
and Ohta (1978) adapted the LBM decou-
pling to binary fluids. Their results differ
from the original LBM theory mainly
wherek
m(t)/k
m(0)1; however, at such
late times neither of these theories is valid,
since they can only account for non-
linearities during the early stages where
k
m(t)/k
m(0)≈1. No theory exists which re-
liably describes the crossover from these
initial time regimes to the late stages,
wherek
m
–1(t)~tis presumed to hold.
An interesting extension is to consider
spinodal decomposition of fluid mixtures
in the presence of flow, e.g., laminar shear
flow (Onuki and Kawasaki, 1978, 1979) or
turbulent flow (Onuki, 1984, 1989c). In
weak shear, the scattering pattern is no
longer the ring pattern familiar from stan-
dard spinodal decomposition, since con-
centration fluctuations become anisotropic.
In polymer mixtures additional effects
arise because the polymer coils become
stretched and oriented by the flow (Pistoor
and Binder, 1988a, b). No such phenomena
are considered here.
T
xxxx
ab ab
aa bb
h
()
()()
xx
xx
xx
−′
−′
⎡
⎣
⎢
+
−′
−′ −′
⎤
⎦
⎥
= (6-65d)
1
8
1
1
3
p| |
d
||
∞
HD=dd
(6-65c)
2∫∫ ′ ∇⋅− ′
×′∇ ′
′
+
′
⎡
⎣
⎢
⎤
⎦
⎥
xx
x
xxx
x
xx
d
d
d
d
d
d
c
c
c
c
F
c
()
() ( )
()
() ()
T
∞∞ ∞
∞= (6-65b)
=d
AHD
A+
−∇+
⎡
⎣
⎢
⎤
⎦
⎥
∫M
cc
F
c
x
xxx
d
d
d
d
d
d() () ()
2
440 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

492. 6.2 General Concepts 441
6.2.7 Effects of Finite Quench Rate
So far both the theory and the discussed
simulations have always assumed instanta-
neous quenches from an initial temperature
T
ito a final temperatureT
f. With regard to
actual experiments, this is extremely ideal-
ized. Often the early stages of spinodal de-
composition have already been passed dur-
ing such a continuous quench where the
temperature is gradually lowered. While it
is believed that the late stages are not af-
fected by the “quenching history”, the lat-
ter can have a drastic effect on both early
and intermediate stages of phase separa-
tion. Unfortunately, in general, the problem
is complicated – the behavior of both the
thermodynamic functions and of the mobil-
ityM(T) in the full regime fromT
itoT
f
may affect the phase separation behavior.
Thus, relatively little theoretical effort has
been devoted to this problem (Houston et
al., 1966; Carmesin et al., 1986). Here we
only quote a few results from the model
calculation of Carmesin et al. (1986), since
the work by Houston et al. (1966) only
considers fluctuations in the initial state
and not in the intermediate states visited in
the quench, which is an approximation that
usually cannot be justified.
As an example, Fig. 6-19 shows a situa-
tion similar to the quench treated in Fig. 6-
8, but here the quench is not carried out in-
stantaneously from infinite temperature to
T/T
c= 4/9, but takes several steps: att=0
the system is cooled instantaneously to
T
1/T
c= 0.75185, att=1toT
2/T
c= 0.67667,
at
t=2 toT
3/T
c= 0.60148, att=3 to
T
4/T
c= 0.5263, and att=4 toT/T
c=4/9,
where the system is later maintained. This
stepwise quenching is more readily acces-
sible to calculation than a fully continuous
quench. A further simplification of the cal-
culation is to neglect any temperature de-
Figure 6-19.Scaled structure
function (left) and normalized
diffusion constantD
˜
eff(q,t)∫
q
2
d[lnS
˜
(q, t)]/dt(right) vs.q
orq
2
, respectively, for the step-
wise quenching procedure de-
scribed in the text. Top, CHC
approximation; bottom, LBM
theory. Ten scaled times
tare
shown. From Carmesin et al.
(1986).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

493. pendence of the mobility. Comparing Fig.
6-19 with Fig. 6-8, characteristic differ-
ences are noted: even in the CHC approxi-
mation there is no longer a unique intersec-
tion point forS(q,
t)–S(q, 0) at different
times
t; apart fromt=0,D
˜
eff(q,t) in the
“Cahn plot” exhibits pronounced curva-
ture, although it depends relatively weakly
on time. Note that in the LBM approxima-
tion we can no longer see any distinct shift
of the maximum position ofS
˜
(q,
t) for the
times shown; this happens because first
(at the intermediate temperature steps)
smaller-qcomponents become more ampli-
fied and later (at the final temperature) am-
plification occurs at larger values ofq. This
behavior just happens to offset the coarsen-
ing tendency. This example shows that care
must be taken in drawing any conclusions
about the validity of the CHC approxima-
tion from experimental data – the effects of
fluctuations in the final state, the gradual
onset of nonlinearities, and finite quench
rate effects are interwoven in a compli-
cated manner, and very accurate measure-
ments and detailed knowledge of the
system parameters are indispensable for
disentangling all these effects.
These effects are much more dramatic
if the mobility has a thermally activated
behavior,M(T)~exp(–E
act/k
BT), with
E
actÔk
BT
f. As a second example, we con-
sider a shallow continuous quench from
T
i= 1.01T
ctoT
f= 0.99T
cduringt= 0.001,
the temperature being lowered linearly
with time (Fig. 6-20). Although this is as
good an approximation for an instantane-
ous quench as is experimentally feasible,
nevertheless, for the choiceE
act/k
BT
c
= 900, more relaxation takes place during
the quench than in the remaining early time
interval (
t&1). The qualitative behavior of
the CHC and LBM approximations is iden-
tical – only the absolute scales of the struc-
ture factors differ significantly.
6.2.8 Interconnected Precipitated
Structure Versus Isolated Droplets,
and the Percolation Transition
Whereas so far the description of the
structures formed by phase separation has
focussed on the behavior of the equal-time
structure factorS(k,t)attimetafter the
quench, we now concentrate on the de-
scription of these structures in real space.
In Fig. 6-2 it was illustrated that during late
or intermediate stages two different pat-
terns of behavior are present, depending
on the volume fraction
Fof the minority
phase. For small
F, the minority phase is
confined to independent clusters well sep-
442 6 Spinodal Decomposition
Figure 6-20.Structure functionS(q, t) vs. scaled
wavevectorqfor scaled times
tas indicated in the
figure, for a continuous quench (linear cooling from
T
i= 1.01T
ctoT
f= 0.99T
cduringt= 0.001), using
E
act/k
BT= 900. Top: CHC approximation; bottom:
LBM approximation. From Carmesin et al. (1986).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

494. 6.2 General Concepts 443
arated from each other; for largerF,we
have a percolating interconnected network
(in viewing Fig. 6-2 we expect an isotropic
system, such as a fluid or glassy mixture; in
crystalline solids the shape of the precipi-
tates reflects the anisotropy of the interfa-
cial free energy between the coexisting
phases, and elastic interactions may even
lead to a regular rather than a random ar-
rangement of these precipitates).
This difference in morphology of the
growing structures often –and erroneously
– is associated with the distinction between
nucleation and growth versus spinodal de-
composition; it is then claimed that the per-
colation of the growing structure is the
hallmark of spinodal decomposition, while
it is assumed that well separated domains
must have been formed by nucleation and
growth. We maintain, however, that the
distinction between nucleation and spino-
dal decomposition, meaningful only for the
earliest stages of phase separation,must
notbe confused with this distinction in
morphology of the precipitated structures,
which is relevant for intermediate and late
stages: even if well separated domains oc-
cur, they may result from a spinodal de-
composition mechanism, and even if a state
decays by nucleation, it may correspond to
an interconnected percolating microscopic
structure of B atoms in the A-rich matrix.
Hence, the transition regime between nu-
cleation and spinodal decomposition and
the line of percolation transitions separat-
ing the regime of finite B clusters (Fig.
6-11a) from the regime where a percolat-
ing cluster occurs which reaches from one
boundary of the system to the opposite one
must in fact cross each other (Fig. 6-21).
Fig. 6-21 shows both the molecular-field
result for the spinodal and the actual re-
gime of gradual transition between nuclea-
tion and nonlinear spinodal decomposition
of a short-range system, defined approxi-
mately by the region fromDF*
MF=10k
BT
toDF*
MF=k
BT, whereDF*
MFis the mean-
field result for the nucleation barrier (this
shift of the transition region from the mo-
lecular-field result towards the binodal was
disregarded in Fig. 6-13). Two different
percolation transition lines are drawn,
which depend on theresolutionwith which
the system is observed. Suppose the resolu-
tion is very fine, such that individual atoms
can be distinguished (this is the situation
envisaged in Fig. 6-11a). Then in the sin-
gle-phase region we encounter a percola-
Figure 6-21.Schematic phase diagram (temperature
vs. concentrationc
–
) of a three-dimensional, short-
range Ising lattice model of a binary mixture. Since
the diagram is symmetric along the linec
–
–c
crit=0.5,
if the meaning of A and B is exchanged, full informa-
tion is given only for the regimec
–
&0.5. Both the
coexistence curve (“binodal”) separating the single-
phase region from the two-phase region and the
mean-field spinodal curve separating metastable
from unstable states in mean-field theory are shown.
The mean-field spinodal is described here by the
equation (c
crit–c)/c
crit=±(1–T/T
c)
1/2
,T
cbeing the
actual critical temperature. The transition regime
from nucleation to spinodal decomposition for a
short-range system, as discussed in the text, is also
indicated. The dash-dotted curves indicate percola-
tion transitions, as discussed in the text. From Hay-
ward et al. (1987).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

495. tion transition of a correlated percolation
problem atc
p
(corr)(T). This line starts out for
TÆ∞at the percolation concentration for
random percolation (c
p
(random)≈0.312 for
the simple cubic lattice (Stauffer, 1985))
and bends over to the left, until it hits the
coexistence curve at the pointT/T
c≈0.96,
c
–
≈0.22 (Müller-Krumbhaar, 1974). This
line continues in the two-phase region as
a transient time-dependent phenomenon,
c
p
(corr)(T,t) (see Fig. 6-22). Hayward et al.
(1987) and Lironis et al. (1989) have
shown that for certain concentrations the
configuration is percolating for a time
intervalt
1<t<t
2, whereas it does not per-
colate for 0&t
1and fort7t
2. If the con-
centration decreases, a critical concentra-
tion is reached wheret
1=t
2(for still lower
concentrations there is no percolation at
all), whereas for increasing concentration
another critical concentration is reached
wheret
2Æ∞. For still higher concentra-
tions the system percolates as the time
exceedst
1and then remains percolating
throughout.
A different behavior occurs if a system
with a much coarser resolution is studied;
at late stages, where the system is phase-
separated on a length scalek
m
–1(t) into the
coexisting phases with concentrations
given by the two branches of the coexis-
tence curve,c
(1)
coex
andc
(2)
coex
, respectively. It
now makes sense to consider percolation
phenomena on much larger length scales
than the lattice spacinga. Suppose we di-
vide our system again into cells of linear
dimensionL(Eq. (6-1)), withaLk
m
–1(t).
Most of these cells will then have concen-
trations close to eitherc
(1)
coex
orc
(2)
coex
.We
may now define clusters consisting of
neighboring cells with concentrations in a
given interval [c
(2)
coex
–dc/2,c
(2)
coex
+dc/2]
and we may ask whether these clusters are
well separated from each other or if they
form an infinite percolating network. Since
l(tÆ∞)Æ∞we may also takeLÆ∞in
this limit and thereforedcÆ0. Hence there
is no longer any ambiguity in this coarse-
grained percolation problem. We expect
that this “macroscopic percolation” will
occur at a critical volume fraction
f
pof the
minority phase, which does not depend on
temperature. Therefore, the line of macro-
scopic percolation concentrations is simply
c
p
(macro)=c
(1)
coex
+(c
(2)
coex
–c
(1)
coex
)f
p
and hence must end in the critical point.
This line is also shown schematically in
Fig. 6-21. Note that this line will be ob-
served experimentally by techniques which
are sensitive to the “contrast” (i.e., differ-
ence in refractive index) between the two
444 6 Spinodal Decomposition
Figure 6-22.Part of the phase diagram of the simple
cubic, nearest-neighbor lattice-gas model, showing
the percolation transition linec
p
(corr)(T,t), where the
timetrefers to an average over the time interval from
t=80 tot= 240 MCS per site during phase separa-
tion. From Hayward et al. (1987).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

496. 6.2 General Concepts 445
coexisting phases, such as observations
with light or electron microscopes.
This phenomenon of “gelation of clus-
ters” atc
p
(corr)(T,t) into an infinite percolat-
ing net has a pronounced effect on the clus-
ter size distributionn
–
l(t), of course, which
was used in the “cluster dynamics” model-
ing of Eq. (6-54). This is shown in Fig.
6-23 wheren
–
l(t) is plotted againstlfor
c
–
= 0.156 in a quench toT= 0, where the
system develops towards a frozen-in clus-
ter size distribution. The curvature ofn
–
l(0)
on the log–log plot reflects the exponential
variation, lnn
–
l(0) ~l, while the straight-
line behavior occurring at later times is
characteristic of the power law of percola-
tion,n
–
l(t)~l
–t
. This percolation problem
is usually disregarded in treatments based
on Eq. (6-54) or related models (Langer
and Schwartz, 1980). Certainly this behav-
ior also makes theories of coarsening,
where the growing droplets are modeled as
essentially spherical, doubtful at volume
fractions
Fclose toF
p; note that from
Fig. 6-22 and the fact that the critical vol-
ume fraction for continuum percolation is
about 0.16 (Scher and Zallen, 1970), we
expect all these theories only to be reliable
for
F0.16. This percolative behavior
n
–
l(t)~l
–t
is in conflict with the scaling be-
havior that is described in the next section,
namely (Binder, 1977, 1989),n
–
l(t)=l
–1
n˜(lt
–d/3
), wheren˜(∫) is a scaling function.
The extent to which this percolation transi-
tion affectsS(k,t) is unclear.
6.2.9 Coarsening and Late Stage Scaling
From Figs. 6-5 and 6-12 it is evident that
the peak positionk
m(t) of the structure fac-
torS(k,t) decreases with increasing time
after the quench. This decrease already re-
flects the onset of a coarsening behavior of
the domains of A-rich and B-rich phases
that have formed (Fig. 6-2). For large
enough times, the domains have grown to a
size much larger than all “microscopic”
lengths (such as the interfacial width).
Then a simple power law should hold,
k
m(t)~t
–x
,tÆ∞ (6-66)
and the structure factor should satisfy a
scaling hypothesis (Binder and Stauffer,
1974, 1976b; Binder et al., 1978; Furu-
kawa, 1978; Marro et al., 1979)
S(k,t)–k
BT/[(∂
2
f/∂c
2
)
T,c
coex
(1, 2)
+r
2
Tk
2
]
~[k
m(t)]
–d
S
˜
{k/k
m(t)} (6-67)
The term subtracted on the left-hand side
of Eq. (6-67) represents the scattering from
concentration fluctuations within the do-
mains, andS
˜
(
z) is a scaling function which
will be discussed later.
Understanding the growth law, Eq. (6-
66), and predicting the associated scaling
functionS
˜
(
z) has been a longstanding
problem that is still not completely solved
(see Bray, 1994, for a recent review). It is
now thought that for solid mixtures, both in
d=2andind= 3 dimensions, but in the ab-
Figure 6-23.Cluster size distribution for a system
ofL
3
lattice site withL= 40. Full symbols give the
cluster size distributionn
–
l(0) corresponding to the
random percolation problem atc
–
= 0.156 at the sim-
ple cubic lattice, and open symbols denote the cluster
size distribution averaged over the time intervals
shown. From Hayward et al. (1987).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

497. sence of any elastic interactions (see Sec.
6.2.10), a result originally derived by Lif-
shitz and Slyozov (1961) and Wagner
(1961) holds
x= 1/3 (6-68)
This LSW theory is essentially a mean-
field theory valid in the limit of zero
volume fraction
Fof the new phase
[
F∫(c
–
–c
(1)
coex
)/(c
(2)
coex
–c
(1)
coex
)] and consid-
ers the cluster size distributionn
–
l(t) that
was discussed in Sec. 6.2.5, showing that
for
F= 0 there exists a solutionn
l(t)=l
–1
n˜(lt
–d/3
), which implies that the mean
cluster “size” (i.e., volume)l
–
(t) scales as
l
–
(t)~t
d/3
fortÆ∞(for a more detailed
outline of this theory, see Wagner et al.,
2001). Thus the cluster linear dimension
scales as [l
–
(t)]
1/d
~t
1/3
, and if we can ex-
tend this result to nonzero
Fwe would ex-
pectk
m(t)~[l
–
(t)]
1/d
and hence Eq. (6-68)
results. However, despite numerous at-
tempts (e.g., Tokuyama and Enomoto,
1993; Akaiwa and Voorhees, 1994) even
the extension of LSW theory to the case of
small
Fis only approximately possible,
and the accuracy of these extensions is
open to doubt (Mazenko and Wickham,
1995). Different power laws for not so late
stages have also been proposed (e.g.,
Binder, 1977, invoking a cluster diffusion
and coagulation mechanism, and To-
kuyama and Enomoto, 1993), and observed
in computer simulations where atomic dif-
fusion is mediated by a single vacancy
moving through the lattice (Fratzl and Pen-
rose, 1994, 1997). Only for deep quenches
where we must take into account that the
mobilityMin Eq. (6-11) is itself concen-
tration-dependent and (almost) vanishing
in the pure phases, diffusion along inter-
faces results in a slower growth law,x=1/4
(Puri et al., 1997).
“Cluster dynamics” approaches such as
Eq. (6-54) (see Binder, 1977; Mirold and
Binder, 1977; and Binder et al., 1978) and
various extensions (e.g., Langer and
Schwartz, 1980; Kampmann and Wagner,
1984; Wagner et al., 2001) incorporate the
LSW growth law, nucleation and coagula-
tion in a phenomenological way. But they
do not take into account the statistical fluc-
tuations and the correlations in the diffu-
sion field around growing clusters.
Eq. (6-68) has now been confirmed by
approaches based on scaling ideas (Furu-
kawa, 1978, 1984, 1985a, 1988; Bray,
1998), renormalization group concepts
(e.g., Lai et al., 1988; Bray, 1990, 1994),
and theories considering fluctuating ran-
dom interfaces (Mazenko, 1994; Mazenko
and Wickham, 1995). However, the most
convincing evidence that Eq. (6-68) is true
both for critical and for off-critical
quenches ind=2 and ind= 3 comes from
computer simulations (Amar et al., 1988;
Gunton et al., 1988; Gawlinski et al., 1989;
Huse, 1986; Rogers et al., 1988; Rogers
and Desai, 1989; Chakrabarti et al., 1993).
We emphasize that Eq. (6-68) holds both
ind=2 and ind= 3 dimensions. The situa-
tion is different for fluid binary mixtures:
the droplet diffusion–coagulation mecha-
nism (also called “Brownian coalescence”)
predicts (Binder and Stauffer, 1974)
k
m(t)~t
–1/d
(6-69)
while mechanisms invoking hydrodynamic
flow of interconnected structures imply
(Siggia, 1979)
k
m(t)~t
–1
,d= 3 (6-70)
andk
m(t)~t
–1/2
,d= 2 (San Miguel et al.,
1985), in the “viscous hydrodynamic re-
gime” (Bray, 1994), whilek
m(t)~t
–2/3
in
the “inertial hydrodynamic regime” (Furu-
kawa, 1985c).
A theoretical problem that is still out-
standing is the calculation of the scaling
functionS
˜
(
z) in Eq. (6-67) (see Bray
446 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

498. 6.2 General Concepts 447
(1994), Puri et al. (1997) and Mazenko
(1994, 1998) for recent discussions). We
mention here only a few approaches very
briefly. Rikvold and Gunton (1982) used
Eq. (6-56), following Binder et al. (1978)
but treating the depletion zones around
growing clusters more realistically. Clearly,
their model is only qualitative. More ambi-
tious early attempts are due to Furukawa
(1984, 1985a, b), Ohta (1984), Tomita
(1984) and Tokuyama et al. (1987). An
equation due to Furukawa (1984) has been
extensively compared with experimental
data on Al–Zn alloys (Komura et al., 1985)
S
˜
(
z)~z
2
/(g˜
2
/2 +z
2+g˜
) (6-71)
where
g˜=d+1 for strongly off-critical
quenches (cluster regime) and
g˜=2dfor
critical quenches (percolative regime).
S
˜
(
z) thus exhibits “Porod’s law” (see e.g.,
Glatter and Kratky (1982))
S(k,t)~k
–(d+1)
(6-72)
in the cluster regime only. Eq. (6-71) also
fails to reproduce the exactly established
(Yeung, 1988) behavior for small
z,
S
˜
(
z)~z
4
. For systems without hydrody-
namics, good empirical forms forS
˜
(
z)
describing the limiting behavior both for
small
zand for largezcorrectly and ac-
counting both for simulation and experi-
ment have been constructed by Fratzl et al.
(1991), but the derivation of such functions
from more fundamental theories is still not
fully solved (Mazenko, 1994, 1998; Ma-
zenko and Wickham, 1995; Bray, 1994).
Fig. 6-27 shows a comparison of some
theoretical predictions for the halfwidth of
the scaling functionS
˜
with corresponding
experiments (Kostorz, 1991). It is intri-
guing to note that the behavior ofS
˜
(
z) does
not change much when the volume fraction
fchanges from the “cluster regime” at
smaller
fthrough a percolation transition
to the interconnected regime at large
f.Re-
call that for fluid binary mixtures even the
exponentxis different in these two regimes
(Eqs. (6-69), (6-70)). For the percolative
regime of fluid mixtures, Furukawa (1984)
has proposed an approximate relation for
q
mvs.t
whereA*andB* are adjustable constants.
Again a more precise treatment has to rely
on numerical calculations (Koga and Ka-
wasaki, 1991, 1993; Puri and Dünweg,
1992; Valls and Farrell, 1993; Shinozaki
and Oono, 1993; Alexander et al., 1993;
Bastea and Lebowitz, 1995).
6.2.10 Effects of Coherent Elastic Misfit
When phase separation occurs on a crys-
talline lattice it is often the case that the
two phases differ slightly in their crystal
structure or their lattice constants, thus in-
troducing elastic strains in the crystal. The
resulting elastic interaction is long-range
and typically anisotropic and may consid-
erably change the phase separation pro-
cess, e.g., in metal alloys (for a recent re-
view, see Fratzl et al., 1999; an extensive
treatment of this problem is also given in
Khachaturyan, 1983).
The coherent misfit strain tensore
0
ij
is
the strain required to transform the (undis-
torted) lattice of one phase into the (undis-
torted) lattice of the other. In metals, parts
of this strain can be relaxed by misfit dislo-
cations disrupting the continuity of the lat-
tices between the two phases. This process
will not be discussed here. In cases of non-
zero misfit strain,e
0
ij
≠0, an influence on
spinodal decomposition is expected if one
of the following conditions is violated:
(i) both phases have the same elastic
stiffness tensor;
( ) [ */ * arctan ( */ * )
arctan */ *] *
qAB BAq
BA B
mm
= (6-73)
−
−
−
1
1
twww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

499. (ii) the misfit strain is purely dilatational;
(iii) the elastic stiffness tensor is isotropic;
(iv) the crystal can be considered infi-
nitely large;
(v) the stress depends linearly on the
strains.
If all five conditions are satisfied, then
phase separation can proceed indepen-
dently of the misfit strains (Bitter–Crum
theorem, see, for example, Cahn and
Larché (1984)). Otherwise, we can expect
changes in the shape of the single-phase
domains, e.g., from spherical to cuboidal or
plate-like shapes but also in their spatial ar-
rangement and coarsening kinetics. Typi-
cally the tendency towards shape changes
increases when single-phase droplets be-
come larger because the elastic energy (be-
ing proportional to the droplet volume) in-
creases faster with the radius than the sur-
face energy.
The introduction of elastic misfit effects
into the theory of spinodal decomposition
means substitutingf
cgbyf
cg+win Eq. (6-
15). The functionwis the elastic energy
density stored in the lattice
(6-74)
where
l
ijmnis the elasticity tensor or stiff-
ness tensor (which may depend on the alloy
composition).
De
ij(x)=e
ij(x)–e
0
ij
(x) (6-75)
is the difference between the strain at posi-
tionxin the elastic equilibrium,e
ij(x), and
the misfit strain,e
0
ij
(x). Inserting this into
the nonlinear Cahn–Hilliard equation (Eq.
(6-14)), we obtain (Larché and Cahn, 1982;
Onuki, 1989a–c)
(6-76)
∂
∂
∇
∂
∂
+
∂
∂
−∇
⎧
⎨
⎩
⎫
⎬
⎭
c
t
M
f
c
w
c
rkT c= cg
B
222
wee
ijmn
ijmn ij mn
=
1
2∑lDD() ()xx
with
(6-77)
The first term arises from a dependence of
the elastic constants on composition and
the second one from the composition de-
pendence of the misfit strain.
The main difficulty in the use of Eq. (6-
77) is the determination ofDe
ij. Since the
strains in the alloy can be assumed to relax
much faster to their equilibrium values
than the concentration profiles,De
ijwill al-
ways be given by the elastic equilibrium
condition
(6-78)
where the stress tensor is defined by
Hooke’s law
t
ij=l
ijmnDe
mn (6-79)
As a consequence, the equilibrium strain at
each point in the material is a (non-local)
functional of the entire concentration pro-
filec(x,t). In the special case when the
elastic constants
l
ijmnare independent of
composition, the elastic problem can be
solved by Fourier transformation (Khacha-
turyan, 1966, 1983) to give
(6-80)
or
(6-81)
which enters the equations (6-76).V
el(u)is
an elastic potential that depends on the
stiffness constants
l
ijmnand the misfit
straine
0
ij
(Khachaturyan, 1983). The aver-∂
∂
−−
∫
w
c
tV ctcy(,) ( )((,) )xxyy=d
el
3
wV ctc
ctc xy
=
dd
el
1
2
33
∫∫ −−
×−
()((,))
((,) )
xy x
y
j
ij
j
t
x
∑
∂
∂
=0
∂
∂
−
∑
∑
w
cc
ee
e
e
c
ijmn
ijmn
ij mn
ijmn
ijmn ij
mn
=
d
d
d
d
1 2
0
l
l
DD
D
448 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

500. 6.2 General Concepts 449
age concentration within the specimen is
calledc
–
.
In the case of isotropic elasticity and
misfit strain, Eq. (6-81) reduces to (Onuki,
1989a– c)
(6-82)
where
h
ais the change of lattice constant
with concentrationc, that is
e
0
ij
(c)=h
a(c–c
–
)d
ij (6-83)
Kand
n
Pare Young’s modulus and the
Poisson coefficient of the elastic matrix,
and
d
ij=0 ifi≠j, d
ii=1 (Kronecker delta).
Eq. (6-82) has been introduced by Cahn
(1961). The consequence of this equation is
a shift of the spinodal line towards lower
temperatures because Eq. (6-44) has to be
replaced by
Hence, the coherent elastic misfit between
the phases may stabilize the solid solution,
even though the demixing into twoinco-
herentlyseparated phases would decrease
the total energy (Cahn, 1961). The reason
is that the coherency condition forces the
alloy to store a considerable amount of
elastic energy in the lattice, which could be
released by creating an incoherent boun-
dary between two regions both having the
(undistorted) equilibrium lattice structure.
For anisotropic stiffness constants
l
ijmn
or anisotropic misfit strainse
0
ij
, the chemi-
cal potential due to elastic interactions,
∂w/∂c, will depend on the direction in the
crystal. This has already been recognized
by Cahn (1962) (see also Khachaturyan
(1983) and Onuki (1989 a–c)). Numerical
solutions of Eq. (6-76) combined with Eq.
(6-81) have shown the development of
strongly anisotropic domains (Nishimori
∂
∂
+
∂
∂
∂
∂
+
−
2
2
2
2
2
2
2
2
1
0
f
c
w
c
f
c
K
a
cg cg
P==
n
h
∂
∂−
−
w
c
K
ctc
a=
P
2
1
2
n
h
[(,) ]x
and Onuki, 1990). A discretized version of similar equations was used by Khachatur- yan and coworkers to study the behavior of single precipitates (Wang et al., 1991) or the evolution of precipitate morphologies (Wang et al., 1992, 1993). Moreover, Monte Carlo simulations of phase separa- tion have also been performed, including elastic misfit interactions between un- equally sized atoms on a lattice (Fratzl and Penrose, 1995, 1996; Laberge et al., 1995, 1997; Lee, 1997, 1998; Gupta et al., 2001). A further approach is the simulation of the coarsening of particles with sharp inter- faces to the matrix, describing the elastic misfit interaction in the framework of mac- roscopic elasticity theory. Recent examples are the studies by Su and Voorhees (1996), Abinandanan and Johnson (1993, 1996) or Jou et al. (1997) (see also the review by Fratzl et al. (1999)). The common observa- tions in all these approaches are that the domains become very anisotropic and typi- cally orient parallel to crystallographic di- rections of the alloy crystal (Fig. 6-24d, e). Moreover, the spatial arrangement of the domains becomes progressively more peri- odic. Despite these enormous changes with respect to the case without elastic misfit interactions, the growth law of a typical do- main size is often still described by the re- sult of the LSW theory (Eq. (6-68)). Ifex-
ternal stressis applied, an additional reor-
ientation of the domains either parallel or perpendicular to the applied stress is ob- served (Laberge et al., 1995, 1997; Weinka- mer et al., 2000). Many of these effects are observed in real alloys, most notably for the technically important nickel-based superal- loys (Maheswari and Ardell, 1993; Conley et al., 1989; Sequeira et al., 1995; Paris et al., 1995, 1997; Fährmann et al., 1995).
If the elastic stiffness constants
l
ijmnde-
pend on alloy composition (see the first term in Eq. (6-77)), this can result in anom-www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

501. alously slow precipitate growth. The most
spectacular observation, however, is that
(at sufficiently late times in the coarsening
process) the softer phase always “wraps”
precipitates of the harder phase (that is, the
phase with the higher stiffness) and, there-
fore, stays percolated even if it is the mi-
nority phase (Onuki and Nishimori, 1991;
Sagui et al., 1994; Leo et al., 1998; Orli-
kowski et al., 1999), see Fig. 6-24a–c.
Most recently, the possibility of atomic
ordering within the precipitates has also
been included in the theory outlined in this
section (Sagui et al., 1994; Wang and
Khachaturyan, 1995), as well as in Monte
Carlo simulations on an elastic lattice (Nie-
laba et al., 1999). This amounts to studying
decomposition close to a tricritical point,
including the effects of elastic misfit inter-
action. With these features, the theoretical
models are now capable of describing a sur-
prising number of details in the evolution of
alloy microstructures (e.g., Li and Chen,
1998). An example is shown in Fig. 6-25.
6.3 Survey of Experimental
Results
The number of experimental studies de-
voted to spinodal decomposition is enor-
mous: first, it is a widespread phenomenon
450 6 Spinodal Decomposition
Figure 6-24.Typical snapshot pictures using the
Cahn–Hilliard equation with elastic interactions (Eq.
(6-77)). (a–c) is the case of isotropic elasticity but
where the elastic stiffness depends on composition.
The stiffer phase is shown white with volume frac-
tions of 0.3, 0.5 and 0.7, respectively (from Onuki
and Nishimori, 1991). Note that the softer phase al-
ways “wraps” stiffer particles. (d, e) is the case
where the elastic stiffness has cubic anisotropy. The
volume fraction of the white phase is 0.5 and 0.7, re-
spectively (from Nishimori and Onuki, 1990).
Figure 6-25.Transmission electron micrograph
taken in the (001)-plane of a Ni–Al–Mo where the lattice spacing between matrix and precipitates (gamma-prime phase) is different by
h
a=–0.5%
(aged for 5 h at 1253 K). Cube-like precipitates can be seen, aligned along the elastically soft directions, [010] and [100]. (b) Results from computer simula- tions of an Ising model with elastic interactions on a simple square lattice and with repulsive interaction of like atoms on nearest-neighbor sites (with interac- tion energyJ) and attractive interaction of like atoms
on next-nearest-neighbor sites (energyJ/2). The dis-
ordered phase (containing mostly A atoms) is shown in black and the ordered phase (consisting of about 50% A and 50% B atoms) is shown in white. The overall concentration of B atoms was 0.35 and the run was performed at a temperature ofT= 0.567J/k
B
on a 128¥128 lattice and stopped after 10
6
Monte
Carlo steps. (c) The same alloy and heat treatment as in (a), but now with an external compressive load of 130 MPa applied along the vertical [010]-direction. (d) The same model, temperature and annealing time as in (b), but with an additional external load along the vertical direction. Experimental data (a and c) are from Paris et al. (1997) and simulation data (b and d) from Nielaba et al. (1999).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

502. 6.3 Survey of Experimental Results 451
that occurs in diverse systems; second,
there has been much interest from a theo-
retical point of view in this phenomenon,
and therefore many experimental studies
have been carried out in an attempt to test
some of the theoretical concepts.
This section cannot present a complete
overview of all these experiments. More
detailed reviews of various aspects of the
experimental work can be found in Haasen
et al. (1984), De Fontaine (1979), Gerold
and Kostorz (1978), Gunton et al. (1983),
Goldburg (1981, 1983), Beysens et al.
(1988), Kostorz (1988, 1994), Hashimoto
(1987, 1988, 1993), and Nose (1987),
among others. Here we attempt only to
give a few representative examples to illus-
trate some of the points discussed in the
theoretical section, and at the same time
show similarities – as well as differences –
between different systems.
6.3.1 Metallic Alloys
Some systems in which spinodal decom-
position has been very extensively studied
are Al–Zn alloys (e.g., Hennion et al.,
1982; Guyot and Simon, 1982, 1988; Si-
mon et al., 1984; Osamura, 1988; Komura
et al., 1985, 1988; Mainville et al., 1997),
ternary Al–Zn–Mg alloys (Komura et al.,
1988; Fratzl and Blaschko, 1988), Ni-
based alloys such as Ni–Al, Ni–Ti, Ni–Cr,
and Ni–Mo (Kostorz, 1988; see also
Kampmann and Wagner, 1984), Al–Li al-
loys (Furusaka et al., 1985, 1986, 1988; Li-
vet and Bloch, 1985; Tomokiyo et al.,
1988; Che et al., 1997; Schmitz et al.,
1994; Hono et al., 1992), Mn–Cu alloys
(Gaulin et al., 1987), Ni–Si alloys (Chen et
al., 1988; Cho and Ardell, 1997), Fe–Cr al-
loys (Katano and Iizumi, 1984), and vari-
ous other ternary alloys, e.g., Cu–Ni–Fe
(Lyon and Simon, 1987; Lopez et al., 1993)
or Ni–Al–Mo (Fährmann et al., 1995;
Paris et al., 1995, 1997; Sequeira et al.,
1995). In all these systems, the structure
factor looks very similar to the early data
on Au–Pt alloys (Singhal et al., 1978) and
those of Al–Zn (Mainville et al., 1997) re-
produced in Fig. 6-5. These results are in
qualitative agreement with the nonlinear
LBM theory and the computer simulations.
The hallmarks of the linearized Cahn the-
ory (exponential increase in intensity with
time, time-independent intersection point
ofS(k,t), maximum positionk
mofS(k,t)
independent of time) are never found. In a
few cases it has been found that at early
timesk
m(t) is nearly independent of time,
e.g., in Mn
67Cu
33(Morii et al., 1988), and
this fact has been taken as evidence for the
validity of the Cahn–Hilliard–Cook (CHC)
approximation. In view of the fact that
k
m(t) almost independent oftcan also re-
sult as an effect of continuous rather than
instantaneous quenching (Sec. 6.2.7) from
a nonlinear theory, we feel that the question
of whether the CHC theory applies to any
of the metallic systems quantitatively is
still unanswered. Certainly we expect that
the CHC theory gives a useful order of
magnitude estimate ofk
m(0) and the initial
growth rateR
m, as it does for the theoreti-
cal models (Fig. 6-5d), provided a spinodal
curve is not close. Experiments by Guyot
and Simon (1982) and Hennion et al.
(1982) were deliberately carried out choos-
ing temperatures both above and below the
spinodal curve of Al–Zn, for several con-
centrations. The structure factor in both
cases is qualitatively the same, as was pre-
dicted (see Sec. 6.2.5). This classical set of
experiments in our view definitely shows
that the spinodal singularity does not play
any role in the unmixing kinetics of Al–Zn
alloys. Note that it is clear that the effective
potential in this alloy is predominantly
short range; however, this alloy does have
a significant misfit of atomic radii betweenwww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

503. Al and Zn, and thus the “coherent phase di-
agram” (where elastic strain fields are not
released) is depressed by about 30 K below
the thermodynamic (incoherent) miscibil-
ity gap, and the spinodal curves are then
similarly shifted, providing an example
of the effects of elastic interactions (Sec.
6.2.10). Despite these significant long-
range elastic forces, the data for Al–Zn
spinodal decomposition are surprisingly
similar to the Monte Carlo results on sim-
ple nearest-neighbor Ising models of al-
loys, where all such elastic effects are not
included. In other cases, such as many bi-
nary and ternary Ni alloys (see, for exam-
ple, Fig. 6-25) elastic misfit interactions
lead to highly anisotropic precipitate mor-
phologies and to ordered arrangements of
precipitates (see Sec. 6.2.10).
There is ample experimental evidence
for the scaling of the structure factor at late
stages (Eq. (6-67)), and the validity of
the Lifshitz–Slyozov exponentx=1/3 (Eq.
(6-68)). Sometimes smaller exponents are
found, e.g.,k
m
–1(t)~t
x
withx≈0.13 to
x≈0.2 (see, for example, Furusaka et al.,
1986; Osamura, 1988), but typically data
extending over only 1.5 decades in time are
available. A general problem is that some
misfit between matrix and precipitates is
present in many real alloys. Moreover, va-
cancy concentrations may be larger than
the equilibrium value after the quench and
then decrease gradually. Both effects could
affect the exponent in real alloys. Katano
and Iizumi (1984), Furusaka et al. (1988)
and Forouhi and De Fontaine (1987) found
a regime of 1.5 decades in time where
x=1/6 whereas after a rather sharp cross-
overx=1/3 at later times. Other experi-
ments (e.g., Morii et al., 1988) where the
k
m
–1(t) vs.trelationship is also measured
over about three decades in time find a
much smoother crossover from the initial
stages, where the log–log plot ofk
m
–1(t)vs.
tis curved and a well-defined exponentx
cannot be identified, to the final Lifshitz–
Slyozov behavior,x=1/3. Katano and Ii-
zumi (1984) interpret the resultx=1/6 in
terms of the mechanism (Binder and Stauf-
fer, 1974) that B-rich clusters in a solid
solution show a random diffusive motion
since B atoms evaporate randomly from the
cluster and re-impinge at another boundary
position, thus leading to a small shift in
the cluster center of gravity. The resulting
cluster diffusivity decreases strongly with
increasing cluster size. Assuming then that
two clusters coalesce when they meet in
their random motions, we arrive atk
m(t)~
t
–1/(d+3)
inddimensions. Although such a
mechanism certainly exists, it is not clear
whether it ever dominates during a well-
defined time interval, since the Lifshitz–
Slyozov–Wagner (1961) mechanism com-
petes with it and should dominate, at least
for long times. Monte Carlo simulations
considering diffusion via vacanies indicate
that cluster–diffusion–coagulation could
be important at low temperatures (Fratzl
and Penrose, 1997). This cluster–diffu-
sion–coagulation mechanism was origi-
nally proposed to explain the correspond-
ing small values of the exponentxseen in
Monte Carlo simulations (Bortz et al.,
1974; Marro et al., 1975). However, more
accurate simulations (Huse, 1986; Amar et
al., 1988) are rather consistent with a grad-
ual approach to the asymptotic Lifshitz–
Slyozov–Wagner (1961) law without an
intermediate regime characterized by a
well-defined different exponentx. In fact,
the data can usually be fitted to an equation
derived by Huse (1986):
k
m
–1(t)=A+Bt
1/3
(6-84a)
while the original treatment of Lifshitz and
Slyozov (1961) yielded
k
m
–3(t)=A¢+B¢t (6-84b)
452 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

504. 6.3 Survey of Experimental Results 453
whereA,B,A¢, andB¢are constants. Eq. (6-
84a) can also be interpreted by an effective
exponent in the growth law,
X
eff(t)∫d{logk
m
–1(t)}/d (logt)
=
1
–
3
[1 –Ak
m(t)]
Evidence for this law is found in various
simulations (Amar et al., 1988; Gunton et
al., 1988) and some experiments (Morii et
al., 1988; Chen et al., 1988; Alkemper et
al., 1999).
There have also been numerous attempts
to characterize experimentally the scaling
functionS
˜
() in Eq. (6-67). Typical data for
Al–Zn alloys are shown in Fig. 6-26. Al-
though the agreement with Eq. (6-71) looks
impressive, the theory (Furukawa, 1984)
neglects anisotropy, and so does the data
analysis on polycrystalline samples of Ko-
mura et al. (1985). Single-crystal work (Si-
mon et al., 1984), however, shows pro-
nounced anisotropy. Anisotropy was also
seen, for example, in scattering from Ni-
based alloys as a result of the coherent
elastic misfit (Kostorz, 1988; Fährmann et
al., 1995; Sequeira et al., 1995). In early
stages of phase separation, the anisotropy
increases roughly linearly with the mean
radius of precipitates,Rµl
1/3
(Paris et al.,
1995). The reason is that the elastic misfit
energy (which favors anisotropy) is pro-
portional to the volume of the precipitate,l,
while the surface energy (which favors
round shapes) depends onl
2/3
.
The scaling function has been deter-
mined experimentally for a number of al-
loys and there is a general trend that the
width of the scaling function decreases
with increasing volume fraction of precipi-
tate phase. A large collection of data is
shown in Fig. 6-27, mostly taken from the
review by Kostorz (1991). This decrease
can be understood qualitative as follows: if
Lis a typical period in the spatial arrange-
Figure 6-26.(a) Scattering cross-section measured
for neutron small-angle scattering from Al–10 at.%
Zn polycrystals at 18 °C (Komura et al., 1985). (b)
Scaling plot of the data shown in (a). Full curve is a
fit to Furukawa’s (1984) function, Eq. (6-71). From
Komura et al. (1985).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

505. ment of the precipitates, the maximum po-
sition of the structure function varies as
L
–1
. For spherical precipitates of radiusR,
the width of the structure function varies
roughly asR
–1
. Hence, the width of the
scaling function behaves approximately as
R/Lµ
F
–1/3
for small volume fractionsF.
A similar argument may also be developed
for larger volume fractions, where the mi-
crostructure corresponds to percolated do-
mains instead of isolated droplets (Fratzl,
1991), leading to the prediction shown in
Fig. 6-27 (broken line).
6.3.2 Glasses, Ceramics,
and Other Solid Materials
Early experimental data on glassy
systems such as the Na
2O–SiO
2system
have been reviewed by Jantzen and Her-
man (1978). Some data have been inter-
preted in the framework of the linearized
theory of spinodal decomposition (e.g.,
Yokota, 1978; Acuna and Craievich, 1979;
Craievich and Olivieri, 1981), but not all
features of the linear theory can be demon-
strated quantitatively in these materials,
and thus no real evidence for the signifi-
cance of a spinodal curve is present. While
the isotropy of these systems is clearly a
simplifying feature, coupling to structural
relaxation sometimes needs to be consid-
ered (Yokota, 1978), which implies a sig-
nificant complication (Binder et al., 1986).
An investigation of late stages was
carried out by Craievich and Sanchez
(1981) for the B
2O
3–PbO(A
2O
3) glass at
T= 0.65T
cwhere the critical point of un-
mixing isT
c≈657 °C. The structure factor
S
˜
(k,t) was obtained in a time range of
about 12 to 400 min, and Eq. (6-66) was
found to be obeyed withk
m
–1(t)~t
x
,
x≈0.23. The early stages were studied by
Stephenson et al. (1991) and were found to
qualitatively agree with the Cahn–Hilli-
ard–Cook theory (Eq. 6-27). Qualitatively,
the behavior is very similar to the results
for metallic alloys (Sec. 6.3.1) and to com-
puter simulations.
At this point, we also mention the appli-
cation of spinodal decomposition of boro-
silicate glass-forming melts to produce
porous Vycor glass; on cooling the melt be-
low its demixing temperature it decom-
poses into an SiO
2-rich phase and a B
2O
3-
alkali-oxide-rich phase. The latter is acid
soluble and can be leached out with suit-
able solvents leaving a fully penetrable mi-
croporous glass. The small-angle scattering
from such materials has been examined and
interpreted in the framework of the theory
of spinodal decomposition (Wiltzius et al.,
1987).
Qualitative observations of phase separ-
ation on a local scale attributed to spinodal
decomposition have been reported for a va-
riety of glassy materials and ceramics. Ex-
amples include rapidly quenched Al
2O
3–
SiO
2–ZrO
8(McPherson, 1987), La–Ni–Al
amorphous alloys (Okamura et al., 1993),
454 6 Spinodal Decomposition
Figure 6-27.Full width at half maximum (FWHM)
of the scaling function normalized such that the posi-
tion of the maximum is located at 1. Black stars are
for Al–Ag alloys (Langmayr et al., 1992), all other
data points are taken from the review by Kostorz
(1991) and correspond to Al–Zn, Pt–Au, Cu–Mn,
Fe–Cr and Al–Li. Full and broken lines indicate pre-
dictions from various models (RG: Rikvold and Gun-
ton (1982), TEK: Tokuyama et al., (1987), FL: Fratzl
and Lebowitz (1989)).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

506. 6.3 Survey of Experimental Results 455
and SiC–AlN ceramics (Kuo et al., 1987).
The crystallization behavior of bulk metal-
lic glasses, such as Zr–Ti–Cu–Ni–Be, has
been found to be strongly influenced by de-
composition (e.g., Wang et al., 1998b).
Of course, spinodal decomposition is
also expected to occur in various nonmetal-
lic mixed crystals, e.g., oxides such as the
TiO
2–SnO
2system (Flevaris, 1987; Taka-
hashi et al., 1988) and semiconducting
GaInAsP epitaxial layers (Cherns et al.,
1988; Mcdevitt et al., 1992). Often these
situations are difficult because of lower
crystal symmetry (the SnO
2–TiO
2system
is tetragonal) and strong elastic lattice mis-
fit effects (Flevaris, 1987).
6.3.3 Fluid Mixtures
Binary fluids containing small mole-
cules such as lutidine–water (Goldburg et
al., 1978; Goldburg, 1981, 1983; Chou and
Goldburg, 1979, 1981) and isobutyric
acid–water (Wong and Knobler, 1978,
1979, 1981; Chou and Goldburg, 1979,
1981) are classic systems where nonlinear
spinodal decomposition was observed via
light scattering techniques. Since the inter-
diffusion in fluid mixtures proceeds much
faster than in solids, we must consider
quenches in a narrow region just belowT
c,
in order to take advantage of critical slow-
ing down. This implies that we must al-
ways work in the non-mean-field critical
region (in the phase diagram in Fig. 6-13),
i.e., nonlinear effects are very strong, and
so it is not expected that the linearized the-
ory of spinodal decomposition will account
for these systems. In addition, very early
stages are not observable because the un-
mixing proceeds too fast. This is best seen
when working with rescaled variables
(Chou and Goldburg, 1979, 1981), defining
q
m(t)∫k
mx(t=0)
and
t=tD(t=0)/ x
2
(t=0)
where
x(t= 0) andD(t= 0) are the correla-
tion length and the interdiffusion constant
in the initial state, respectively. Early times
then mean a
tof the order of unity. Fig. 6-
28 shows that only scaled times
tÊ6 are
accessible, whereas earlier times are ac-
cessible for fluid polymer mixtures (see
Sec. 6.3.4). The growth rate exponentx
Figure 6-28.Log–log
plot ofq
mvs.tfound in
the polymer mixture poly-
styrene (PS)–poly(vinyl
methyl ether) (PVME)
(data labelled system K),
compared with isobutyric
acid–water (l) and 2,6-luti-
dine–water mixtures (m).
Systems (K, m) are data
from Chou and Goldburg
(1979). From Snyder and
Meakin (1983).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

507. (q
m
–1(t)~t
x
) crosses over fromxnear
x≈1/3 for
t=10 tox≈1 at very late times,
as expected (Siggia, 1979) (see Eq. (6-70)).
The behaviorq
m
–1(t)~t
–1/3
, which for off-
critical quenches where the structure corre-
sponds to well separated clusters is be-
lieved to be the true asymptotic behavior
for
tÆ∞(Siggia, 1979), is attributed to a
cluster diffusion and coagulation mecha-
nism (Binder and Stauffer, 1974; Siggia,
1979). Note that for fluid droplets the
diffusion constants decrease in proportion
to their radii with increasing droplet size,
due to Stokes’ law, and hence the cluster
diffusion–coagulation mechanism never
becomes negligible in comparison with
the Lifshitz–Slyozov (1961) evaporation–
condensation mechanism.
Fig. 6-28 shows, however, that the cross-
over to the asymptotic power law is grad-
ual. This smooth behavior is approximately
described by the nonlinear theory of spino-
dal decomposition of Kawasaki and Ohta
(1978). However, this theory can give only
a poor account of the full structure func-
tion. This is not surprising, since it is only
expected to be accurate for
tÁ10 (see Sec.
6.2.6). Again, the problem of calculating
the scaled structure functionS
˜
(∫) (Eq. (6-
67)) arises and is as difficult as in the case
of solids (Furukawa, 1985a).
A problem for binary fluids is the effect
of gravity. Since the two coexisting phases
usually differ in density, one phase must go
to the top and the other to the bottom of the
container. This effect (for a more detailed
discussion see Beysens et al., (1988)) im-
plies a smearing of the critical region for
the unmixing critical point of a sample of
finite height, and also affects the late stages
of phase separation (where even hydrody-
namic instabilities may set in; see Chan
and Goldburg (1987)). The effect of gravity,
however, can be strongly reduced if iso-
density systems are used (using mixtures of
methanol and partially deuterated cyclo-
hexane, a perfect density matching atT
cis
possible; see Houessou et al., (1985)) or
space experiments are performed, where
the gravitational effect can be reduced by a
factor of 10
4
. Beysens et al. (1988) showed
that such experiments agree with the re-
sults obtained from isodensity systems. By
such means not only can the accuracy of
data of the sort shown in Fig. 6-28 be sub-
stantially improved, but several decades
can be added to a plot ofq
mvs.t(Fig. 6-
29). The full curve in this plot is the func-
tion (Eq. (6-73)) proposed by Furukawa
(1984), which reduces to Eq. (6-70) for
large
t, i.e., the result of Siggia (1979).
More recently, a growth of the mean drop-
let size with the power-law exponent 1/3
could be followed over more than seven
decades in a microgravity experiment (Per-
rot et al., 1994).
Interesting extensions involve spinodal
decomposition in fluid mixtures under
weak steady-state shear (Chan et al., 1988;
Krall et al., 1992; Lauger et al., 1995;
Hashimoto et al., 1995; Hobbie et al.,
1996) or periodically applied shear (Bey-
sens and Perrot, 1984; Joshua et al., 1985),
as well as in strongly stirred mixtures
where a turbulent suppression of spinodal
decomposition occurs (Pine et al., 1984;
Chan et al., 1987; Easwar, 1992). Many of
these observations can be explained on the
basis of the theoretical considerations of
Onuki (1984, 1986a, 1989c).
Finally, we draw attention to phase sep-
aration phenomena in more exotic fluids
such as surfactant micellar solutions (Wil-
coxon et al., 1988, 1995), polymer solu-
tions near their
Q-point, e.g., polystyrene
in methyl acetate (Chu, 1988), gel–gel
transitions (such a transition occurs, for ex-
ample, in N-isopropylacrylamide gel with
water as a solvent, where this system phase
separates into two states with different sol-
456 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

508. 6.3 Survey of Experimental Results 457
vent concentrations and hence a different
swelling ratio of the gel; see Hirotsu and
Kaneki (1988)), and lipid membranes with
dissolved protein.
6.3.4 Polymer Mixtures
As pointed out in Secs. 6.2.4 and 6.2.5,
mixtures of long flexible macromolecules
are particularly well suited model systems
for the study of phase separation kinetics:
(i) owing to the mutual entanglement of the
random polymer coils, their interdiffusion
is very slow for high molecular weights, so
the early stages can be studied; (ii) polymer
mixtures exhibit a well-defined mean-field
critical regime, where the linear theory of
spinodal decomposition should hold, and a
spinodal curve can be defined; (iii) chang-
ing the molecular weights while other
parameters (in particular, intermolecular
forces!) remain constant allows a more
stringent test of theories than for other
systems. For example, it has been possible
recently to obtain by laser scanning confo-
cal microscopy (Jinnai et al., 1997) three-
dimensional images of phase-separated
polymer blends that look very similar to
pictures obtained from theory (Fig. 6.30).
Therefore, it is gratifying that quantita-
tive agreement with the linear theory of
spinodal decomposition was obtained in
careful experiments with various poly-
mer blends, e.g., polystyrene–poly(vinyl
methyl ether) (Okada and Han, 1986; Han
et al., 1988; Sato and Han, 1988), polybu-
Figure 6-29.Behavior of the scaled wavevectorq
m(t) versus scaled timet. Full curve is from Furukawa
(1985), data points refer to mixtures of methanol and cyclohexane in space-flight experiments (F) or to isoden-
sity mixtures of methanol and partially deuterated cyclohexane for various quench depths, as indicated. The
late-time behavior, whereq
m(t)~t
–1
is observed, is magnified in the inset. From Beysens et al. (1988).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

509. tadiene and styrene–butadiene random co-
polymer mixtures (Hashimoto and Izumi-
tani, 1985), and mixtures of deuterated and
protonated polybuta-1,4-diene (Bates and
Wiltzius, 1989), among others. As an ex-
ample, Fig. 6-31 now provides an experi-
mental counterpart to Figs. 6-15, 6-16,
which were obtained from simulation. Note
that here Eq. (6-23) is used, i.e., fluctua-
tions in the final state are neglected – there-
fore the Cahn plot bends over for large
wavevectors, andR(q)/q
2
never becomes
negative, owing to the second term on the
right-hand side of Eq. (6-29). Moreover, by
carrying out quenches to different tempera-
tures it is shown that bothq
2
m
(0) and the ef-
fective diffusion constantD
0vary linearly
with (1–T/T
c), as expected from Sec. 6.2.5.
The onset of nonlinear effects can be iden-
tified, but it is not accurately described by
LBM-type theories (Sato and Han, 1988).
458 6 Spinodal Decomposition
Figure 6-30.Three-dimensional representation of a
bicontinuous phase-separated polymer blend. Inves-
tigation of the blend by laser scanning confocal mi-
croscopy gives results very similar to numerical solu-
tions of the nonlinear theory of spinodal decomposi-
tion (Jinnai et al., 1997).
Figure 6-31.(a) Early stage growth of concentration
fluctuations in a mixture of protonated and deuter- ated polybuta-1,4-diene, with polymerization index N
H= 3180,N
D= 3550, at critical volume fraction
f
D= 0.486 andT= 322 K (T
c= 334.5 K). Four repre-
sentative scattering vectors are shown. From the straight-line fits to these semi-logarithmic plots the amplification factorR(q) is extracted, which is
shown as a Cahn plot in (b). From Bates and Wiltzius (1989).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

510. 6.3 Survey of Experimental Results 459
At intermediate time a behavior
k
m
–1(t)~t
x
is observed, wherexvaries with
quench depth. This behavior is not under-
stood theoretically. In this regime of times,
nonlinear effects gradually become impor-
tant. A transition stage then occurs, before
the full scaling behavior of the structure
factor is observed in the final stage. The
scaling plot forq
m(t) vs.tis presented in
Fig. 6-32, and the scaling function for the
structure factor is shown in Fig. 6-33. It is
seen that Furukawa’s (1984) function gives
a reasonable overall account of the data.
The fact that the curves for cyclohexane
methanol and for poly(vinyl methyl
ether)–polystyrene (PVME–PS) mixtures
are offset is not considered to be very seri-
ous, because there are some uncertainties
involved in the rescaling ofkandtin order
to obtain the scaled variablesqand
t; only
in terms of the scaled variables does it
make sense to compare different materials
with each other and to discuss whether a
truly universal behavior exists.
A possible interpretation of the devia-
tions from universality in the “intermedi-
ate” and “transition” stage in Fig. 6-32 is
the fact that Bates and Wiltzius (1989) con-
sider deep quenches, for which 1–T/T
cis
no longer very small. Only for 1–T/T
c1
can we expect a truly universal behavior. In
any case, comparison of Fig. 6-29 and Fig.
6-32 with Fig. 6-28 clearly demonstrates
the experimental progress made during less
than a decade; the accuracy of the data in
Fig. 6-32 is substantially better and the
range of
tis nearly one decade larger.
Figure 6-33 shows the behavior of the
scaling function at small and at largefor a
polymer blend (a) and for a metal alloy (b).
This behavior is of the form
x
withx=–4
at large. At small,xis close to 4 in both
polymers and metal alloys. The decay at
largefollows directly from Porod’s law
(Porod, 1951) which is a consequence of
sharp interfaces between the single-phase
domains. The exponentx= 4 at smallhas
been proposed by Yeung (1988) from an
analysis of the Cahn–Hilliard equation. Up
to now, there is no theoretical expression
for the scaling function that is capable of
reproducing all the features in Fig. 6-33.
The model of Fratzl and Lebowitz (1989)
uses the correct exponents at lowand at
largeand predicts a volume fraction de-
Figure 6-32.Log–log plot ofq
m(t
–
) versus scaled
time
t
–
for the polymer mixtures in Fig. 6-28 (dif-
ferent symbols represent different samples) at
T= 298 K. Solid curve is a fit of Furukawa’s (1984)
function, Eq. (6-73), to the data in the “final stage
region”. The poly(vinyl methyl ether)–polystyrene
(PVME–PS) curve represents results of Hashimoto
et al. (1986), and the curve for the cyclohex-
ane–methanol small-molecule mixture is taken from
data of Guenoun et al. (1987). The crossover time
from the intermediate stage to the transition stage is
denoted as
t
–
s, from the transition stage to the final
stage
t
–
w. Note that the scaled timet
–
for the small-
molecule mixtures is defined differently, since
D
–
eff=D
0is not well defined. From Bates and Wilt-
zius (1989).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

511. pendence in reasonable agreement with ex-
periments (Fig. 6-27), but it fails to repro-
duce the shoulder in the scaling function
appearing on the right side of the maximum
for polymers (Fig. 6-33a) and even some
metal alloys (Fig. 6-33b). Earlier theories
also fail to produce this shoulder, except
for the approach by Ohta and Nozaki
(1989), which predicts a function qualita-
tively similar to the data in Fig. 6-33a, but
is still quantitatively off.
6.4 Extensions
Basic aspects of the theory of spinodal
decomposition were treated in Sec. 6.2.
Only the generic phase diagram (Figs. 6-
3b, 6-13, and 6-21) of a binary mixture
with a miscibility gap has been considered.
In this section, we briefly mention the re-
lated phenomenon of ordering kinetics
(Fig. 6-3a) and also discuss cases where
formation of order and unmixing compete.
The effects of certain complications (im-
purities, effects of finite size and free sur-
faces) are briefly assessed.
6.4.1 Systems Near a Tricritical Point
If a binary system exhibits a line of crit-
ical pointsT
c(c
–
) for an order–disorder
transition, a tricritical pointT
t(c
–
t) may oc-
cur where this critical line stops, and below
this tricritical point both ordering and
phase separations occur simultaneously
(Griffiths, 1970; Lawrie and Sarbach,
1984). Two order parameters,
y(x) and
c(x), are simultaneously needed to de-
scribe such phenomena. Examples of this
situation occur in some magnetic alloys
where the order–disorder transition de-
scribes a magnetic ordering,
3
He–
4
He mix-
tures (here
ydescribes the superfluid order
parameter and hence must be taken as a
460 6 Spinodal Decomposition
Figure 6-33.(a) Log–log plot for the structure-fac-
tor scaling function in the late stages of a quench to
313 K for the polymer mixture in Fig. 6-31. The solid
curve is the theoretical prediction of Ohta and No-
zaki (1989). The straight lineI(q)~q
–4
demonstrates
the validity of Porod’s law. The inset shows low-q
behavior, as measured in the intermediate stage,
where a power lawI(q)~q
4.5
is found. From Bates
and Wiltzius (1989). (b) Similar plot for the scaling
function in the metallic alloy Al–15 at.% Ag (Lang-
mayr et al., 1992). The full line corresponds to the
model function of Fratzl and Lebowitz (1989) for a
volume fraction of
F= 0.27. This functionS
–
() in-
creases with
4
at smalland decreases with
–4
at
large(Porod’s law).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

512. 6.4 Extensions 461
complex variable), and also crystallo-
graphic order–disorder phase transitions in
alloys and in adsorbed layers at surfaces
are known to exhibit such tricritical points.
In alloys, the first discussions of phase sep-
aration in tricritical systems come from Al-
len and Cahn (1976, 1979a, b) in the con-
text of Fe–Al alloys. An elegant extension
of the linearized theory of spinodal decom-
position to tricritical
3
He–
4
He mixtures
was given by Hohenberg and Nelson
(1979). Nonlinear effects have mostly been
studied for the model with simple relaxa-
tional dynamics of the order parameter
(Dee et al., 1981; San Miguel et al., 1981).
Here we briefly mention the case of
3
He–
4
He mixtures, since the most detailed
experiments exist for these systems (Hoffer
et al., 1980; Sinha and Hoffer, 1981; Benda
et al. 1981, 1982; Alpern et al., 1982).
3
He–
4
He mixtures have no practical appli-
cations in materials science, but we men-
tion them here as a model system for the
study of phase separation near a tricriti-
cal point. The concepts developed (and
checked experimentally) for this model
system can be carried over to more com-
plex materials such as ternary mixtures,
magnetic alloys, etc. Instead of Eqs. (6-4)
and (6-8), the expression for the free en-
ergy functional is now
wherer
0,u,v, c
n
–1,l
0, andgare pheno-
menological coefficients that should de-
pend only on temperature and not on the lo-
cal
3
He concentrationc, which is measured
D
|| ||
|| | |
||
||∫
kT
ru
r
cc c
lc
B
n
=d
(6-85)∫
⎧
⎨
⎩
+
++∇
++−−+
+∇
⎫
⎬
⎭
−
x
x
x
1
2
1
4
1
6
1
2
1
2
1
2
0
24
62 2
12 2
34 0
0
22yy
yy
cgymm
()
()
()
v
D
from some reference value that fixes the
constant
D
0, andm
3andm
4are the chemi-
cal potentials of
3
He and
4
He, respectively.
For fixedc,T, and
D=m
3–m
4+D
0the
static equilibrium follows from seeking the
minimum of
(6-86)
wherer˜=r
0+2gDc
n,u˜=u–2 c
ng
2
,and
c=
c
n[D–g|y|
2
] was eliminated from the
equation. Foru˜> 0 there is a second-order
transition from normal fluid to superfluid
atr˜= 0, whereas foru˜< 0 the transition is
of first order. The critical point occurs for
u˜=0.
Whereas for the binary mixture a single
Langevin equation for the concentration
fieldc(x,t)and
y(x,t) had to be derived
(Eq. (6-15)), here we need two equations.
In fact, the entropy densityS(x,t) must
also be added (Hohenberg and Nelson,
1979), and thus the calculation becomes
complicated. However, since
yis not a con-
served quantity, it relaxes much more rap-
idly, hence the assumption is made that it al-
ways adjusts to the local equilibrium corre-
sponding to the local concentrationc(x,t).
Instead of the simple exponential function
in Eq. (6-23), additional oscillating terms
due to the “second-sound” mode are found:
S(k,t)=a
1e
2R(k)t
+a
2e
[R(k)–D 2k
2
]t
(6-87)
¥cos (u
2kt)+a
3e
–2D 2k
2
t
cos (2u
2kt)+a
0
wherea
0,a
1,a
2, anda
3are constants,u
2is
the second-sound velocity andD
2its damp-
ing coefficient. In the spinodal region,R(k)
is positive, and not too close to the tricriti-
cal point for smallk,R(k)–D
2k
2
should
also be positive. Therefore in addition to
the first term on the right-hand side of Eq.
(6-87), which is the analog of Eq. (6-23),
there is another exponentially growing
term oscillating in time. This would imply
Fr u=
n
1
2
1
4
1
6
1
2
2462
˜˜|| || ||yyyc+++ v Dwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

513. a “flickering” component in the scattering.
However, the linearized theory of spinodal
decomposition is not expected to hold for
3
He–
4
He mixtures, since neither of the
interaction rangesrandl
0in Eq. (6-85) are
expected to be large. Note, however, that
the Ginzburg (1960) criterion for tricritical
systems shows that mean-field theory is es-
sentially correct for tricritical points; hence
nonlinear effects do not become worse
whenTÆT
t, and unlike in Eqs. (6-51) and
(6-52), we expect no factor depending on
the temperature distance fromT
tunder the
conditions of validity of the linear theory.
For tricritical systems, both experiments
and simulations (Sahni and Gunton, 1980;
Sahni et al., 1982; Ohta et al., 1988; Sagui
et al., 1994; Wang et al., 1998 a; Gorents-
veig et al., 1997; Nielaba et al., 1999) have
shown a nonlinear behavior for all times
accessible to study, and a scaling behavior
of the structure factor as in Eq. (6-67) again
applies to the late stages.
6.4.2 Spontaneous Growth of Ordered
Domains out of Initially Disordered
Configurations
There are many examples where the dy-
namics of a phase change involve the spon-
taneous formation of ordered structures.
Suppose an alloy such asb-CuZn, which is
a prototype realization of Ising-model type
ordering (Als-Nielsen, 1976), is quenched
fromT
0>T
ctoT<T
c(Fig. 6-3a). Then
again the initial homogeneously disordered
state is thermodynamically unstable, and
we expect that ordered domains will form.
Since no sign of the order parameter is pre-
ferred, and there is no symmetry-breaking
field, the average order parameter
y
–
in the
system,
y
–
= (1/V )∫dx
y(x,t), will remain
y
–
= 0: domains with both signs of the order
parameter form equally often. Again, the
(unfavorable) domain wall energy acts as a
driving force leading to coarsening of the
domain structure, similar to the coarsening
discussed for spinodal decomposition (Sec.
6.2.9).
The description of the initial stages of
this domain growth process closely par-
allels the treatment presented in Eqs. (6-4)
to (6-26). Let us assume a free energy func-
tional similar to Eq. (6-85), with pheno-
menological constantsr
0,u,v, andr:
and we emphasize that the order parameter
yis not a conserved quantity. Thus instead
of Eqs. (6-10) and (6-11) we write:
(6-89)
describing a simple relaxational approach
towards equilibrium, with
Gbeing an ap-
propriate rate constant. Again, fluctuations
must be added to this equation as in Eqs.
(6-14) to (6-17), so that the final result is
where the random force
h
T(x,t) now satis-
fies the relationship
·
h
T(x,t)h
T(x¢,t¢)Ò
=·
h
2
T
Ò
Td(x–x¢)d(t–t¢)
·
h
2
T
Ò
T=2k
BTG (6-91)
Again the first step of the analysis is a lin-
earization approximation, similar to Eqs.
(6-18) to (6-26). For the structure factor,
which we now define as follows,
∂
∂
∂
∂
⎡
⎣
⎢
⎤
⎦
⎥
⎧
⎨
⎪
⎩
⎪
−∇
}+
y y
y
yh(,) [(,)]
(,) (,)
x x
xx
t
t
ft
rkT t t
T
T
=( 6-9)
cg
B
G 0
22
∂
∂
−y
m
myy(,)
˜(,)
˜(,) ( (,))/ (,)
x
x
xxx
t
t
t
ttt
=
=
G
dD d∫
D∫
kT
ru
r
B
= d (6-88)∫
⎧
⎨
⎩
+
++∇
⎫
⎬
⎭
x
x
1
2
1
4
1
6
1
2
0
24
62 2yy
yy
[()]v
462 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

514. 6.4 Extensions 463
S(k,t)=·d y
–k(t)dy
k(t)Ò
d
y
k(t)∫∫dx[exp(ik·x)] y(x,t) (6-92)
the result is analogous to Eq. (6-23) (De
Fontaine and Cook, 1971):
S(k,t)=S
T0
(k)exp[2R(k)t] (6-93)
with
(6-94)
Note thatkin Eqs. (6-92) to (6-94) de-
scribes the distance in reciprocal space
from the superstructure Bragg spotQ
Bcor-
responding to the considered ordering.
Suppose nowf
cghas the form assumed
in Eq. (6-88), withu>0,v>0, and
r
0=r¢(T–T
0) changes sign atT
0(this is the
standard Landau description of a second-
order transition; see Stanley (1971) or
Binder (2000)). Then
(∂
2
f
cg/∂y
2
)
y
–=r
0+12uy
–2
+30vy
–4
is equal tor
0< 0 fory
–
= 0, i.e., the initial
configuration is always unstable, and all
modes
y
k(t) grow in the interval from
0<k<k
c, withk
cstill being given by Eq.
(6-25). However, Eq. (6-26) does not hold
here and the maximum growth rate occurs
fork= 0, as is obvious from Eq. (6-94).
In a case whereu< 0, however, the phase
transition does not occur forT=T
0where
r
0changes sign, but rather at a higher
temperatureT
c=T
0+3u
2
/32r¢v(Binder,
1987a). ForT
0<T<T
cthe state withy
–
=0
is metastable, and the ordering reaction
again needs a nucleation process (Chan,
1977; Fredrickson and Binder, 1989). The
temperatureT
0hence again plays the role
of a spinodal point (“ordering spinodal”;
De Fontaine, 1979). Such spinodals for
thermally driven first-order transitions
have been calculated for diverse systems,
from alloys such as Cu–Au systems (De
R
f
rkTk
T
()
,
k≡−
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
G
2
2
0
22cg
=
B
y
y
Fontaine and Kikuchi, 1978) to the meso-
phase separation transition of block copol-
ymers (Fredrickson and Binder, 1989). It is
important to realize, however, that the “or-
dering spinodal” is also a mean-field con-
cept, suffering from the same “ill-defined-
ness” as the unmixing spinodal. Taking
into account both statistical fluctuations
and nonlinear effects, as they are both con-
tained in Eqs. (6-90) and (6-91), we again
expect to find a smooth transition between
the nucleation of order and the “spinodal
ordering” mechanism (Cook et al., 1969),
as described by Eqs. (6-93) and (6-94). In
addition, nonlinear effects will limit the
predicted exponential growth, as they do in
the case of spinodal decomposition.
This similarity of the behavior also car-
ries over to the late stages of growth, where
a scaling similar to Eq. (6-67) applies: but
nowk
m(t) denotes the half-width of the
peak (unlike spinodal decomposition, the
maximum growth always occurs at the
Bragg positionQ
Bdescribing the order,
i.e.,k∫0). Since the order parameter
yin
Eq. (6-90) is not conserved, a faster growth
law than Eq. (6-68) is predicted (Lifshitz,
1962; Allen and Cahn, 1979b; Ohta et al.,
1982; Bray, 1994):
k
m
–1(t)~t
1/2
(6-95)
In a binary alloy undergoing an order–dis-
order transition, or in a layer adsorbed at a
surface at constant coverage, the conserva-
tion of concentration (or density, respec-
tively) may change the growth law (Eq. (6-
95)): Sadiq and Binder (1984) suggested
that the excess concentration (or excess
density, respectively) contained in certain
types of domain walls could again lead to a
mechanism of the Lifshitz–Slyozov type,
i.e., Eq. (6-68). The validity of this sugges-
tion is not yet confirmed (Binder and Heer-
mann, 1985; Binder et al., 1987). Figs.
6-34 and 6-35 give some examples of thewww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

515. behavior found in computer simulations of
a square lattice model of a two-dimen-
sional alloy, where there is a repulsive
interaction between AB pairs on both near-
est- and next-nearest-neighbor sites. For
c
–
= 0.5 the ordering is the two-component
(2¥1) structure: there are four kinds of do-
mains, where in the ground state full rows
of A atoms alternate with full rows of B
atoms, and these rows may be oriented in
either thex-direction ory-direction for a
square lattice. Fig. 6-34 shows the growth
of these ordered domains in the simulation
of a quenching experiment for this model,
and Fig. 6-35 the resulting growth of the
superstructure Bragg intensity. The scaling
behavior of Eq. (6-67) is well verified.
Similar results have also been found for
simulations of various other models (see
Binder et al., (1987), Furukawa (1985a),
Bray (1994), for review.
Experiments on the kinetics of the for-
mation of ordered superstructures have
been carried out both for two-dimensional
monolayers adsorbed at surfaces (Wang
and Lu, 1983; Wu et al., 1983; 1989; Trin-
gides et al., 1986, 1987; Henzler and
Busch, 1990), three-dimensional metallic
alloys (e.g., Hashimoto et al., 1978; Nishi-
hara et al., 1982; Noda et al., 1984; Takeda
et al., 1987; Katano and Iizumi, 1988; Ko-
nishi and Noda, 1988), dielectric materials
such as K
2Ba(NO
2)
4(Noda, 1988) or
K
2ZnCl
4(Mashiyama, 1988), martensitic
materials such as KD
3(SeO
3)
2(Yagi and Lu,
464 6 Spinodal Decomposition
Figure 6-34.“Snapshot pictures” of the computer simulation of the ordering process of a 120¥120 square lat-
tice model of an alloy with repulsive interactions between A–B pairs on nearest- and next-nearest-neighbor
sites,
e
nn=e
nnn. A quench from a disordered configuration at infinite temperature tok
BT/e
nn= 1.33 is performed
forc
–
= 0.5 (a second-order transition to the (2¥1) structure occurs in equilibrium atk
BT
c/e
nn≈2.1). Time evo-
lution occurs via random nearest-neighbor A–B exchanges (i.e., the model of Kawasaki (1972)). Times shown
refer to 10 MCS after the quench (upper part), 200 MCS (middle) and 1700 MCS (bottom). Only B atoms are
shown (using four different symbols to indicate whether a B atom belongs to a domain of type 1, 2, 3, or 4, see
Fig. 6-35a), A atoms are not shown. From Sadiq and Binder (1984).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

516. 6.4 Extensions 465
1988), graphite intercalation compounds
such as transition metal chlorides in graph-
ite (Matsuura, 1988), molecular crystals
suchasCN
1–xCl
x-adamantane (Descamps
and Caucheteux, 1988), etc. Of course,
there are also cases where phase separation
and ordering occur simultaneously, a clas-
sic example being Fe–Al alloys where
Fe
3Al domains precipitate (Allen, 1977;
Oki et al., 1973; Eguchi et al., 1984).
Again, no attempt at completeness is made
here, we simply quote a number of exam-
ples to emphasize the universality of the
phenomenon.
As expected from the above discussion,
we must distinguish the cases where the
“parent phase” from which the ordered
structure forms is metastable or unstable;
in the latter case, the structure forms by
“spinodal ordering” (De Fontaine, 1979)
whereas in the former case it forms by nu-
cleation and growth, although again this
distinction is not very sharp. An example
of the latter case is Mg
3In, where the order
is of Cu
3Au type but the transition is very
strongly of first order (Konishi and Noda,
1988). The volume fraction of the ordered
phase is then found to obey the well-known
Avrami (1939), Johnson–Mehl (1939)
equation
F(t)=1–exp[–(t/ t)
n
] (6-96)
wherenis a constant characteristic of the
type of nucleation process and
ta time
constant that depends on the degree of
Figure 6-35.(a) Four types of domains (1, 2, 3, and 4) in the (2¥1) structure of a binary alloy AB on the square
lattice (B atoms are indicated by black circles and A atoms by white circles). (b) Structure factorS(q,t)of
superlattice Bragg scattering, at timestafter the quench. Since the lattice spacing is set to unity, the Bragg po-
sitions of the (2¥1) structure are (±p,0)and(0,±p). Only the variation withq
xnearq
x=pis shown. Times are
measured in units of Monte Carlo steps per site. Data are for a computer simulation of a quench fromT=•to
k
BT/e
nn= 1.33 as in Fig. 6-34. (c) Scaling plot of the data shown in (b), with the structure factorS(q,t) being
rescaled with the maximum intensityS(p,t) and the relative distanceq/p–1 from the Bragg position being re-
scaled with the half-widths
s(t) of the peaks in (b). From Sadiq and Binder (1984).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

517. undercooling of the transition. Here we are
more interested in the “spinodal ordering”
as occurs in Cu
3Pd (Takeda et al., 1987),
Ni
3Mn (Katano and Iizumi, 1988), Cu
3Au
(Hashimoto et al., 1978; Nishihara et al.,
1982; Noda et al., 1984), and K
2Ba(NO
2)
4
(Noda, 1988).
Fig. 6-36 shows examples of growing
Bragg peaks, i.e. the experimental counter-
parts to simulation data such as shown
in Fig. 6-35. Whereas for K
2Ba(NO
2)
4we
find a lawk
m
–1(t)~t
1/4
(Noda, 1988), for
Ni
3Mn a crossover fromk
m
–1(t)~t
1/4
at
short times tok
m
–1(t)~t
1/2
at later times is
found (Katano and Iizumi, 1988), and for
AuCu
3alawk
m
–1(t)~t
1/2
(i.e., Eq. (6-95))
is established (Noda et al., 1984). Whereas
thet
1/2
law is expected from various theo-
ries (Lifshitz, 1962; Allen and Cahn,
1979b; Ohta et al., 1982), at
1/4
law occur-
ring over a transient period of time can per-
haps be attributed to the “softness” of the
domain walls between the growing ordered
regions (Mouritsen, 1986; Mouritsen and
Praestgaard, 1988). An interesting obser-
vation of Lifshitz–Slyozov type coarsen-
ing of the structure factor is also reported
for colloid crystallization (Schätzel and
Ackerson, 1993).
In the two dimensional case, Wu et al.
(1989) measured a growth exponentx=
0.28 ± 0.05 for O on W (110). Since the
p(2¥1) phase is believed to have an eight-
fold ground-state degeneracy in this sys-
tem, whereas the theories mentioned above
refer to a two-fold degenerate ordering
only, the interpretation of this result is not
obvious. We note, however, that reasonable
scaling of the structure factor is observed
(Fig. 6-37). For the system silver on Ge
(111), a growth exponentx=1/2 is found
(Henzler and Busch, 1990) and also good
scaling of the structure factor is seen. In
this system, on the other hand, a crossover
to slower growth occurs at later times; the
reason for this behavior is not clear – per-
haps it is due to pinning of domain walls at
defects (see Sec. 6.4.4). At this point, we
note that a similar slowing down of the
growth wherek
m
–1(t) basically stops grow-
ing further, has also been seen in Ni
3Mn
(Katano and Iizumi, 1988) and in the phase
separation of mixtures of flexible and
semi-rigid polymers (Hasegawa et al.,
1988). In neither of these systems is the
very slow growth at late stages understood.
466 6 Spinodal Decomposition
Figure 6-36.(a) Time evolution of the 211 super-
lattice peak of Ni
3Mn annealed at 470 °C for times
up to 34 h. From Katano and Iizumi (1988). (b) Time
evolution of the (–
1
2
,x,–
5
2
) superlattice peak of
K
2Ba(NO
2)
4annealed at 190 K. From Noda (1988).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

518. 6.4 Extensions 467
In domain growth studies of O on W (112)
Zuo et al. (1989) also found scaling behav-
ior at early times but slowing of domain
growth attributed to random field effects at
later times.
6.4.3 Phase Separation in Reduced
Geometry and Near Surfaces
Various cases of reduced geometry may
be of interest; for example, spinodal de-
composition of fluid mixtures confined to
pores in porous media like Vycor glass, or
spinodal decomposition or ordering of thin
films or adsorbed monolayers at surfaces,
and phase separation in small particles or
grains in inhomogeneous materials. All
these problems have seen much recent ac-
tivity, and lack of space allows us to give
only a very brief introduction.
In a porous medium (or gel), a binary
mixture at critical composition experiences
a random chemical potential difference (De
Gennes, 1984). Using concepts from the
random field Ising model (Villain, 1985;
Huse, 1987) we expect both a change in the
phase diagram and slow relaxation (Gold-
burg, 1988; Goldburg et al., 1995; Falicov
and Berker, 1995). However, there is also
the view that we must rather focus on the
wetting behavior on the walls inside a
straight cylindrical pore (Liu et al., 1990;
Monette et al., 1992; Tanaka, 1993; Zhang
Figure 6-37.(a) Dynamic scaling in the growth of
Bragg peaks for W (110) p (2¥1) – O at coverage
F= 0.5 andT= 297 K. From Wu et al. (1989). (b)
Scaled LEED intensities of the÷
-
3peakofAgad-
sorbed on Ge (111) atT= 111 °C. From Henzler and
Busch (1990).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

519. and Chakrabarti, 1994, 1995). Of course,
ultimately we should combine this single-
pore behavior with effects due to the ran-
dom structure of the pore network.
From Sec. 6.2 it should be clear that the
behavior in bulk two and bulk three dimen-
sions is qualitatively similar, although non-
linear phenomena are usually more impor-
tant ford= 2 than ford= 3. In particular,
for polymer mixtures different coils can
interpenetrate each other ford= 3 but not
ford= 2; therefore a mean-field critical re-
gime does not exist for two-dimensional
polymer mixtures.
For thin films that are many atomic di-
ameters thick, there is an interesting inter-
play between finite size effects (Binder et
al., 1987) and surface effects. Usually al-
ready in equilibrium at the surface of a
mixture, the concentration of one compo-
nent of a mixture will be enhanced in com-
parison with the bulk concentration (“sur-
face enrichment” occurs because the inter-
actions between a wall and the two compo-
nents A and B differ from each other). At
the coexistence curve, this preference of
the surface for, say, the B component may
lead to the formation of a wetting layer at
the surface (Dietrich, 1988). Inside the
two-phase coexistence region, surfaces
may hence have a profound effect on the
kinetics of phase separation: surface-di-
rected concentration waves (with wavevec-
tor normal to the surface plane) dominate
the initial growth in the region near the sur-
face (first observed by Jones et al. (1991)
for a polymer mixture). In later stages of
the coarsening, we have an interesting
competition between the domain growth in
the bulk and the possible propagation of a
wetting layer from the surface into the bulk
(see Krausch, 1995; Puri and Frisch, 1997;
and Binder, 1998, for recent reviews).
6.4.4 Effects of Quenched Impurities;
Vacancies; and Electrical Resistivity
of Metallic Alloys Undergoing Phase
Changes
In this subsection, we draw attention to a
variety of topics that cannot be discussed
here in any depth, owing to lack of space.
(i) Quenched Impurities
Throughout this chapter, only ideal
systems have been considered where phase
separation is triggered by spontaneous
fluctuations. However, real materials al-
ways contain impurities, whereas in fluids
these impurities are mobile (“annealed
defects”) and hence act like a dilute addi-
tional component of a multi-component
mixture, in solids such impurities are often
immobile (“quenched defects”) at the tem-
peratures of interest. This frozen-in disor-
der may have two effects: in a metastable
region, free energy barriers for nucleation
are reduced and henceheterogeneous nu-
cleationis facilitated (Zettlemoyer, 1969).
In the late stages of spinodal decomposi-
tion or domain growth, there is an impor-
tant interaction between such defects and
domain walls. Whereas so far it has been
assumed that domain walls may diffuse
freely owing to their capillary wave excita-
tions, the randomly spaced defects act like
a random potential in which the interface
moves. Since thermal activation is now
required to overcome the barriers of this
potential, at low temperatures the domain
wall motion is dramatically slowed owing
to such impurities. Simulations of this
problem have been carried out by Grest and
Srolovitz (1985) and Srolovitz and Grest
(1985). More work has been devoted to the
related problem of the dynamics of the ran-
dom field Ising model (see Villain, 1985;
and Nattermann, 1998, for reviews). In all
these systems, we expect that the growth
law of the ideal system (Eqs. (6-68) or
468 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

520. 6.4 Extensions 469
(6-95)) will hold until some characteristic
lengthl
cis reached, which depends on the
concentrationc
iof the quenched impurities
(l
c~c
i
–1/dinddimensions). At timestsig-
nificantly exceeding the time given by the
conditionk
m
–1(t)l
c≈1, a logarithmic growth
law (Villain, 1985) is expected:
k
m
–1(t)~lnt (6-97)
A similar crossover in the relaxation from a
fast to a slow growth law may also be
caused by interface pinning at extended de-
fects (e.g., dislocations, grain boundaries)
rather than point defects.
(ii) Vacancies
In solid mixtures, vacancies V are cru-
cial for a microscopic description of inter-
diffusion, which occurs via a vacancy
mechanism rather than by direct A–B ex-
change (Flynn, 1972; Manning, 1986). Mi-
croscopically we therefore need concentra-
tionsc
i
A(t),c
i
B(t), andc
i
V(t) for A atoms, B
atoms, and vacancies as dynamic variables,
respectively, rather than the single concen-
tration variablec
ior concentration field
c(x,t) (Eq. (6-1)). Sincec
i
V(t)1, it may
be possible to reduce the problem to Eq.
(6-15), where the mobilityMthen needs to
be related to the jump rates
G
AandG
Bof
A and B atoms to vacant sites (and the va-
cancy concentration). However, owing to
correlation effects in the vacancy motion,
this is a difficult problem even in the non-
interacting case (Kehr et al., 1989). There
Eqs. (6-6) and (6-14) would yield an inter-
diffusion coefficientD
int=M/[c(1–c)],
i.e.,Mcan be found ifD
intcan be simply
related to
G
AandG
Band the average va-
cancy concentrationc
–V
. AlthoughMobvi-
ously is proportional toc
–V
for smallc
–V
,
computer simulations (Kehr et al., 1989)
show that the simple relationships pro-
posed in the literature to relateMto
G
Aand
G
Bare inaccurate.
A simulation of the early stages of spino-
dal decomposition in such an ABV model
(Yaldram and Binder, 1991) shows that the
general features of the structure factor
S(k,t) are almost the same as those of the
direct-exchange AB model (Fig. 6-5), and
the two models can be approximately
mapped onto each other by adjusting the
time scales. In real systems, however, the
behavior may be more complicated; the va-
cancy concentration does not need to re-
main constant, and it may be that many va-
cancies are created during the quench
which later are annealed out by migration
to surfaces or recombination with intersti-
tials. If this happens, the effective mobility
Mwould itself depend on the timetafter
the quench. Also the vacancy concentra-
tions in equilibrium may be different in A-
rich and B-rich domains, or may become
preferentially enriched in domain boundar-
ies. This did not occur in the simulation of
Yaldram and Binder (1991), since there the
static properties were assumed with perfect
symmetry between A and B and there was
no energy parameter associated with the
vacancies, but this model is still a gross
over-simplification of reality. Fratzl and
Penrose (1994, 1997) found that vacancies
may speed up the coarsening by changing
the mechanism. Note that for very low va-
cancy content but very late stages a faster
growth law (x=1/2) due to cluster–cluster
aggregation has been suggested (Mukher-
jee and Cooper, 1998). These problems
may confuse the proper interpretation of
real experiments.
(iii) Electrical Resistivity of Metallic
Alloys Undergoing Phase Changes
In metallic alloys the quasi-free elec-
trons responsible for electrical conduction
are scattered from the atomic disorder. In
the framework of the Born approximation,
treating the scattering as elastic, the excesswww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

521. resistivityD rdue to concentration fluctua-
tions can be represented as a convolution
of the structure factor·d
y
–k(t)dy
k(t)Òand
the Fourier transform of the effective po-
tential that the quasi-free electrons feel,
namely (Binder and Stauffer, 1976b),
(6-98)
whereeis the charge of an electron,m
effits
effective mass,Nis the number of atoms
per cm
3
,Z
AandZ
Bare the atomic numbers
of the two constituents,äis Planck’s con-
stant, and
kdescribes the screening of the
Coulomb interaction. In Eq. (6-98) it is as-
sumed that the Fermi sphere (whose radius
isk
F) lies completely within the first Bril-
louin zone, and that there is no sublattice
ordering.
Using the structure factor from the com-
puter simulations of Marro et al. (1975),
Binder and Stauffer (1976b) showed that
during spinodal decomposition a resistivity
maximum occurs. This can be understood
since in Eq. (6-98) the main contribution
comes from the largek(kof order
kor of
orderk
F). As the coarsening proceeds,
however, the structure factor is large only
at much smaller values ofk. Scaling con-
siderations (Eq. (6-67)) suggest that
D
r(t)–Dr(∞)~k
m(t)~t
–1/3
This treatment can again be generalized for
the kinetics of ordering alloys. We refer to
Binder and Stauffer (1976a) for a discus-
sion of this problem, and of previous theo-
ries and related early experiments.
Just as an alternative description for con-
centration inhomogeneities in terms of
“concentration waves” and their mean-
square amplitudeS(k,t) in terms of the
D
p
dd dr
a
k=
d
B A eff
eff F
F
N
eZ Z m
nk
k
k
ctct c
kk
222
23 43
02
222
2
4
1
()
()
(/)
[()() ()]
−
×
+
×〈 〉−
<<
−
∫
∫
k
k
kk
time-dependent cluster distribution func- tion (see Sec. 6.2.5), the problem of the electrical resistivity can be rephrased in terms of electron scattering from clusters (Hillel et al., 1975; Edwards and Hillel, 1977). Experimental resistivity data for Al–Zn alloys can be interpreted qualita- tively along such lines (Luiggi et al., 1980).
At this point, we also mention that elec-
trical currents have been found to affect the phase separation behavior of Al–Si alloys (Onodera and Hirano, 1986, 1988) and other alloys. At present, the explanation of these phenomena is unclear.
6.4.5 Further Related Phenomena
As a first point of this subsection, we
draw attention to peculiar morphologies of
structures that may form via phase separa-
tion in fluid mixtures with very strong
dynamic asymmetry, e.g., solution of poly-
mers which are frozen into a glass-like
structure at high density. The resulting vis-
coelastic phase separation leads to the for-
mation of sponge-like patterns (Tanaka,
1994, 1996; Taniguchi and Onuki, 1996) or
rigid foams (Hikmet et al., 1988).
Another recent topic of interest is phase
separation induced by temperature gra-
dients (Kumaki et al., 1996) and velocity
gradients, i.e. shear (Onuki et al., 1997).
The effect of shear flow on the unmixing of
fluids is in fact three-fold, as recently re-
viewed by Onuki (1997): the phase dia-
gram is modified (shear-induced mixing
due to a depression of the critical tempera-
ture may occur), the rheology of the un-
mixed two-phase states is profoundly al-
tered, and the morphology of the patterns
that form is changed.
Finally we mention the interplay of
phase separation and chemical reactions
(Huberman, 1976; Glotzer et al., 1995; Le-
fever et al., 1995). Chemical reactions may
470 6 Spinodal Decompositionwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

522. 6.5 Discussion 471
terminate the coarsening process and thus
freeze in an inhomogeneous pattern.
6.5 Discussion
The understanding of the dynamics of
phase changes of materials has advanced
considerably in recent years; the theoreti-
cal results now present a fairly clear picture
of at least some basic questions, and also
quantitatively reliable experiments are
available for many systems. This chapter
has emphasized the theoretical aspects; its
main conclusions can be summarized as
follows:
(i) The linearized theory of spinodal de-
composition holds only for mean-field type
systems (with long ranges of interactions)
or in systems that are equivalent to them,
such as polymer mixtures.
(ii) Whereas a spinodal curve is well de-
fined in the mean-field limit or for polymer
mixtures in the limit of chain lengthNÆ∞,
in all other cases the spinodal is an ill-de-
fined concept. This implies that the transi-
tion from nucleation to spinodal decompo-
sition is gradual. For polymer mixtures, the
width of this transition region may be nar-
row, as estimations from Ginzburg criteria
show. Experiments are still needed to
check the latter point, whereas the gradual
transition from nucleation to spinodal de-
composition in metallic alloys over a broad
temperature region has been established
experimentally.
(iii) Nonlinear spinodal decomposition
during the early stages can be successfully
described qualitatively by the Langer–
Baron–Miller theory (for solid mixtures)
and the Kawasaki–Ohta theory (for fluid
mixtures). Recent experiments, however,
show that these theories are not quantita-
tively accurate. More work is required to
understand the crossover from these early
stages to the late stage of phase separation
where the structure factor develops to-
wards a scaling limit. It is not clear under
which conditions power-law behavior
k
m(t)~t
–x
can be observed at intermediate
stages, and what the appropriate interpreta-
tion of the exponentxis.
(iv) In the late stages the structure factor
obeys the scaling behavior first suggested
by Binder and Stauffer (1974, 1976b),
S(k,t)~[k
m(t)]
–d
S
˜
{k/k
m(t)}
While the scaling functionS
˜
(∫) proposed
by Furukawa (1984) seems to account well
for the general shape of experimental data
on solid or polymer mixtures, several de-
tails are not reproduced correctly. Good ev-
idence for Porod’s law,S
˜
(∫)~∫
–(d+1)
for
∫1, has been provided. The width ofS
˜
(∫)
increases when the volume fraction
fof
precipitate phase decreases from that of the
critical mixture (Fratzl and Lebowitz,
1989). There we find, as a function of tem-
perature and
f, both morphologies, inter-
connected domains and well separated
droplets. Such a transition can also be ob-
served as a function of time in computer
simulations (Hayward et al., 1987; Lironis
et al., 1989) and experiments on polymer
mixtures (Hasegawa et al., 1988) (see Fig.
6-38).
(v) There is now agreement that in solid
mixtures the behavior at late times is given
by the Lifshitz–Slyozov law,k
m
–1(t)~t
1/3
,
even in the percolative regime (Amar et al.,
1988). This has recently also been verified
in a solid–liquid system that satisfies all
assumptions of the LSW theory (Alkemper
et al., 1999). For critical quenches of fluid
mixtures at late times, Siggia’s (1979) re-
sultk
m
–1(t)~tholds, but further research is
needed to achieve an understanding of the
exponents for the growth of ordered struc-
tures in cases with a higher degeneracy ofwww.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

523. the ordered phase, and to understand how
these growth laws are affected by defects.
Other questions concern the suggestion
(Langer et al., 1975; Binder, 1977) that by
expressingk
mandtas suitably scaled
quantitiesq
mandt, e.g.,q
m(t)=k
m(t)x
andt=D
0t/x
2
, wherexis the correlation
length at phase coexistence andD
0the
interdiffusion constant, a material-inde-
pendent universal function should be ob-
tained (which must depend, however, on
the volume fraction of the phases coexist-
ing in the final equilibrium state). A coun-
ter-example for this universality seems to
be the “Nbranching” found for polymer
mixtures (Onuki, 1986b; Hashimoto,
1988), and more work is needed to under-
stand this problem. Clearly such a univer-
sality, if it exists, would be useful as it al-
lows the phase separation behavior of other
materials to be predicted if properties such
as
xandD
0are known. Of course, we must
distinguish between different “universality
classes” (Binder, 1977) for solids and fluid
mixtures (cf. Figs. 6-28 and 6-39).
(vi) Apart from crude “cluster dynam-
ics” models, there is no theoretical ap-
472 6 Spinodal Decomposition
Figure 6-38.Polarizing optical microscope images
obtained from the same area of a cast film of a binary
mixture of poly(ethylene terephthalate) and a copoly-
ester composed of 60 mol% p-oxybenzoate and
40 mol% ethylene terephthalate at 50% relative con-
centration during isothermal heat treatment at
270 °C. Times at which the images are taken are indi-
cated (in seconds). From Hasegawa et al. (1988).
Figure 6-39.Comparison ofq
mvs.
tbehavior found in metallic alloys
(a)–(h) and in organic glasses (i), (j). Here the scaling is done differently to that in the text, namelyq
m=k
m(t)/k
m(0)
and
t=–2tR[k
m(0)]. Systems used are
Au–60 at.% Pt at 550 °C (a) (Singhal et al., 1978), Al–6.8 at.% Zn atT=108°C
(b), 116 °C (c), 129 °C (d) (Laslaz et al., 1977), Al–5.3 at.% Zn atT= 20 °C (e),
Al–6.8 at.% Zn at 20 °C (f), 90 °C (g), 110 °C (h) (Hennion et al., 1982), 76B
2O
3–19PbO–5Al
2O
3at 450 °C (i)
(Zarzycki and Naudin, 1969), and 13.2Na
2O–SiO
2at 560 °C (j) (Tomo-
zawa et al., 1970). From Synder and Meakin (1983).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

524. 6.6 Acknowledgements 473
proach to describe the behavior in the tran-
sition region from spinodal decomposition
to nucleation. This problem, and theories
connecting consistently the early-time spi-
nodal decomposition to the scaling behav-
ior at late stages, still needs more research.
So far much progress has been due to large-
scale computer simulation, e.g., Amar et al.
(1988) (see Figs. 6-40 and 6-41); it is still a
challenge to explain quantitatively simula-
tion data such as shown in these figures
or experimental data as shown in Fig. 6-32
by analytical theories.
(vii) Finally, we draw attention to the
fact that the concepts developed in the
present context can also be extended to
very different physical phenomena. For
example, the Lifshitz–Slyozov mechanism
can be identified to describe processes such
as the healing of rough surfaces at low tem-
perature (Villain, 1986; Selke, 1987).
6.6 Acknowledgements
This work has profited considerably
from a longstanding and stimulating inter-
action of K. B. with D. Stauffer and D. W.
Heermann. This author thanks them and
also other co-workers, namely C. Billotet,
H.-O. Carmesin, H. L. Frisch, J. D. Gun-
ton, S. Hayward, J. Jäckle, M. H. Kalos, K.
Kaski, K. W. Kehr, J. L. Lebowitz, G. Liro-
nis, A. Milchev, P. Mirold, S. Reulein, A.
Sariban, and K. Yaldram, for fruitful col-
laboration.
The other author (P. F.) is particularly in-
debted to O. Penrose and J. L. Lebowitz for
a longstanding and fruitful collaboration.
Furthermore he is indebted to H. Gupta, C.
A. Laberge, F. Langmayr, P. Nielaba, O.
Paris, G. Vogl, and R. Weinkamer for fruit-
ful interactions.
Figure 6-40.“Snapshot pictures” of the two-dimen-
sional nearest-neighbor Ising model of a phase-sep-
arating mixture evolving after a critical quench to
T=0.5T
cat timest= 5000 MCS (a),t=10
5
MCS (b)
andt=9.8¥10
5
MCS (c). B atoms are shown
in black, A atoms are not shown. Data are for a
512¥512 lattice. From Amar et al. (1988).
Figure 6-41.Log–log plot of the characteristic lin-
ear dimensionsR
G(t)andR
E(t) at timetafter the
quench, for the simulation shown in Fig. 6-40.R
G(t)
is extracted from the correlation function andR
E(t)
from the energy relaxation. The full curve is a fit to Eq. (6-84a). From Amar et al. (1988).www.iran-mavad.com
+ s e l ∂'4 , kp e r i ∂&s ! 9 j+ N 0 e

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480 6 Spinodal Decompositionwww.iran-mavad.com
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532. 7 Transformations Involving Interfacial Diffusion
Gary R. Purdy
Department of Materials Science and Engineering, McMaster University, Hamilton,
Ontario, Canada
Yves J. M. Bréchet
Laboratoire de Thermodynamique et Physico-Chimie Métallurgiques, E.N.S.E.E.G.,
Institut National Polytechnique de Grenoble, France
List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 482
7.1 Introduction................................ 484
7.2 Equilibrium Properties of Solid–Solid Interfaces............ 484
7.2.1 Structures of Grain Boundaries . . . . . . . . . . . . . . . . . . . . . . . 484
7.2.2 Structures of Interphase Boundaries . . . . . . . . . . . . . . . . . . . . 486
7.2.3 Segregation to an Interface . . . . . . . . . . . . . . . . . . . . . . . . . 489
7.3 Interfacial Diffusion............................ 490
7.4 Forces for Interface Migration...................... 492
7.4.1 Capillary Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
7.4.2 Chemical Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
7.4.3 Mechanical Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
7.4.3.1 Elastic Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
7.4.3.2 Plastic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
7.4.4 Frictional Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
7.4.4.1 Solute Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
7.4.4.2 Particle Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
7.5 Examples.................................. 498
7.5.1 The Massive Transformation . . . . . . . . . . . . . . . . . . . . . . . . 498
7.5.2 Chemically-Induced Grain Boundary Migration . . . . . . . . . . . . . . 503
7.5.3 Discontinuous Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . 504
7.5.3.1 Initiation of Discontinuous Precipitation . . . . . . . . . . . . . . . . . . 505
7.5.3.2 Theories of Steady Cooperative Growth . . . . . . . . . . . . . . . . . . 507
7.5.3.3 Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . . 510
7.5.4 Interface Migration in Multilayers . . . . . . . . . . . . . . . . . . . . . 514
7.6 Conclusions................................. 516
7.7 References................................. 516
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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533. List of Symbols and Abbreviations
A interfacial area
b magnitude of the Burgers vector
C
0 average concentration
C
a concentration
DC local difference from the average concentration C
D diffusion constant
D
V/v ratio of the volume diffusion constant and the interfacial velocity
D
–
b
average grain boundary diffusion coefficient
dE/dx measure of force between solute atoms and boundary
DF
ch molar Helmholtz energy change accompanying complete transformation
DF¢
ch molar Helmholtz energy difference across an interface
DF
0
ch
fraction of the total available Helmholtz energy dissipated in volume diffusion
f(C) Helmholtz energy density functional
f
0 measure of curvature of Helmholtz energy–composition relationship
g
B shear modulus
H 3
k
a/(3k
a+4g
B)
J
B flux of solute atoms
K=dA/dVinterface curvature
L half-thickness of a foil specimen
L
a
0
, L
b
0
regular solution parameters for the crystal and boundary phases
l correlation length
M mobility
N
V number of atoms per unit volume
n
a
A
mole number
p
ch interfacial driving force/chemical force
p
h residual chemical force derived from coherency strain
p
i solute drag force
p
W virtual mechanical force
r
1, r
2 radii of curvature
r
i radius of facet
S spacing between precipitate lamellae
s equilibrium ratio of solute concentration in a grain boundary and that in adja-
cent crystal
T/T
m reduced temperature
T
m melting temperature
v boundary velocity
V
f volume fraction of a precipitate
V
m molar volume
V
m
b molar volume of the boundary phase
W applied load per unit area
x, y, z coordinates
Y elastic constant
Z applied tensile stress
482 7 Transformations Involving Interfacial Diffusionwww.iran-mavad.com
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534. List of Symbols and Abbreviations 483
a crystalline phase
a, b included, external phase
g
a activity coefficient
G
B surface excess concentration of solute B per unit area
d boundary thickness
e misfit
e
a interaction parameter; a measure of the non-ideality of the asolid solution
h coefficient of lattice parameter change with composition
q angle of tilt
k gradient energy coefficient
k
a bulk modulus of precipitate
l spacing of parallel edge dislocations
m
A, m
B solvent, solute chemical potentials
W alignment parameter
s specific interfacial Helmholtz energy
s
A, s
B specific grain boundary Helmholtz energy in pure A, B
s
ab specific interfacial Helmholtz energy of a/binterface
j phase field parameter
y order parameter
b.c.c. body-centered cubic
CIGM chemically induced grain boundary migration
DP discontinuous precipitation
DC discontinuous coarsening
f.c.c. face-centered cubic
h.c.p. hexagonal close-packed
STEM scanning transmission electron microscopywww.iran-mavad.com
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535. 7.1 Introduction
Many microstructure-determining reac-
tions in solids are similar in the sense that
they occur primarily at an interface, which
traverses a volume of solid material, and
leaves that volume altered structurally or
chemically, or both. Indeed, most solid–
solid transformations fall into this set; in
this chapter, we will restrict discussion in
the following ways:
1) We will consider only those processes
whose main characteristics are determined
by diffusion within the transformation
front, which may be a grain boundary or an
interphase boundary.
2) We will further restrict attention to
processes that begin mainly at crystal
boundaries, and that lead to heterogene-
ously transformed regions. This restriction
implies that the transformation interface
will be of relatively high energy, and there-
fore that the transformation will often be
initiated with some difficulty.
The subset defined above excludes those
homogeneous transformations which are
initiated everywhere within a parent crystal
(Chap. 5; Wagner et al., 2001), and truly
diffusionless transformations (Chap. 9; De-
laey, 2001). However, the massive trans-
formation survives, as do the processes of
peritectoid precipitation, chemically in-
duced grain boundary migration, and dis-
continuous precipitation. The latter two
(and similar diffusional processes) are
the subject of an excellent monograph by
Pawlowski and Zie¸ba (1991), which is, un-
fortunately, not widely available. It has re-
cently been observed that the study of these
and similar phenomena may be advanced
through the use of artificially prepared
multilayered structures (Klinger et al.,
1997b, c, 1998), which may be subject, for
example, to discontinuous homogenization
or to interfacial precipitation. Before dis-
cussing these processes, a brief review will
be given of the equilibrium properties of
transformation interfaces and of the dy-
namics of interfacial diffusion and inter-
face migration.
7.2 Equilibrium Properties
of Solid–Solid Interfaces
7.2.1 Structures of Grain Boundaries
Grain boundaries constitute a well stud-
ied class of solid state defects which are
structural in nature, and which depend for
their existence on a misorientation between
otherwise identical crystals. If we restrict
attention to macroscopic variables only,
the properties of a planar boundary are ex-
pected to depend on five geometric vari-
ables, three of which define the relative or-
ientations of the two crystals, and a further
two are required to specify the orientation
of the boundary plane. Three additional de-
grees of freedom, associated with sublat-
tice translations, exist on the microscopic
scale appropriate to the modeling of grain
boundary structure.
Much of what is known or inferred about
grain boundary structure is obtained from
simulations using molecular statics or mo-
lecular dynamics, supported in some cases
by high resolution microscopic techniques
(Smith, 1986).
The three main modeling techniques that
have been used for surfaces, grain bound-
aries and (to a lesser extent) interphase
boundaries, are energy relaxation methods,
molecular dynamics, and Monte Carlo sim-
ulations (Sutton and Balluffi, 1995). The
information expected from these simula-
tions concerns both the structural aspects
(molecular relaxation, molecular dynam-
ics), chemical aspects such as segregation
484 7 Transformations Involving Interfacial Diffusionwww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

536. 7.2 Equilibrium Properties of Solid – Solid Interfaces 485
(Monte Carlo), and dynamic aspects such
as diffusion (molecular dynamics). The
structural information stemming from
these methods is very much dependent on
the interatomic potential chosen, and espe-
cially on the anharmonic part, since the
structure of the interface is usually rather
distorted. The dynamic aspects of inter-
faces include the interface diffusivity and
the interface mobility. While the diffusivity
is accessible to these simulations (via mo-
lecular dynamics), the mobility is more of a
problem, because the system size is typi-
cally constrained by available computa-
tional resources. This imposes, for exam-
ple, very large curvatures as driving forces,
and results in high interface velocities
which might not be realistic reflections of
experimental situations (Sutton, 1995).
The chemical aspects accessible to
Monte Carlo simulation also possess two
facets: the thermodynamic equilibrium to
be attained and the kinetics required to
reach it. Provided that the interatomic po-
tential is reliable, the Monte Carlo method
gives an accurate description of chemical
segregation (Treglia and Legrand, 1998).
However, the simulation of the kinetics
needed to reach this state encounters a new
problem of coupling the bulk diffusion to
the diffusion close to the interface. Monte
Carlo simulations with time residence al-
gorithms (Martin et al., 1998) have allowed
the inclusion of the role of vacancies in
bulk diffusion, but the treatment of vacan-
cies in solids with free surfaces or internal
interfaces remains a source of problems
(Delage, 1998).
The whole field of surface and interface
thermodynamics and kinetics of metallic
alloys with tendencies to ordering or un-
mixing has profited from Monte Carlo
simulations. The next step (to introduce
simultaneous structural relaxations and
chemical aspects, e.g., via off-lattice
Monte Carlo methods) still requires further
development.
A surprisingly large amount of our infor-
mation relates to a rather special kind of
interface, the symmetric tilt grain bound-
ary. As suggested by Fig. 7-1, low angle
boundaries are made up of arrays of dislo-
cations, and the low angle symmetric tilt
boundary is composed of parallel edge dis-
locations, whose spacing
lis given by
(7-1)
where bis the magnitude of the Burgers
vector, and
qthe angle of tilt. Studies by
Krakow et al. (1986) suggest that this de-
scription in terms of individual disloca-
tions can be extrapolated, with some mod-
ification, to high angles, e.g., to 26.5°.
More generally, the grain boundary region
is expected to possess the following char-
acteristics:
a) The high angle grain boundary is thin,
perhaps two or three atomic diameters in
width. To a first approximation, it may be
considered a high energy “phase”, con-
strained to have constant volume.
b) The high angle grain boundary is peri-
odic in structure, like its low angle counter-
part. The geometric repeating units are not
lq=b/ sin( / )22
Figure 7-1.A symmetric low angle tilt grain bound-
ary. The tilt angle
qand the edge dislocation spacing
lare related by Eq. (7-1).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

537. lattice dislocations, but are capable of de-
scription in terms of nearly perfect com-
pact polyhedra (Ashby et al., 1979). Depar-
tures from symmetric orientations are often
accommodated by mixing two or more
types of structural units within the bound-
ary. It is the view of Sutton and Balluffi
(1987) that no general criterion for low
grain boundary energy can be developed on
the basis of simple geometry.
c) Metastable states, and even degener-
ate states of different structure but identical
energy, may be formed by the relative mi-
croscopic translation of one of the crystals
(Vitek, 1984).
The computed structures are of great
conceptual value, yielding information that
is inaccessible or accessible only with dif-
ficulty through experiment; they should be
treated with a measure of caution however,
as the computed results depend to some ex-
tent on the choice of potential. Further,
many (but not all) of these simulations are
performed for a temperature of 0 K. Never-
theless, in combination with experiment,
they form a valuable tool in the study of
this complex field.
7.2.2 Structures of Interphase Boundaries
These internal surfaces, unlike grain
boundaries, are necessary elements of an
equilibrium multiphase system; they are
not defects. They separate phases which in
general may differ both chemically and
structurally from one another.
The description of an interface using
continuum concepts dates from the seminal
papers by Cahn and Hilliard (1958) and Al-
len and Cahn (1977). The Cahn–Hilliard
model of a stationary coherent interphase
boundary is illustrated in Fig. 7-2. The
Cahn–Hilliard and Allen–Cahn treatments
deal with two broad cases involving con-
served and non-conserved order parame-
ters. The first applies to chemically hetero-
geneous solids, and the second to structu-
rally heterogeneous materials. The first al-
lows us to derive a diffusion equation for the
evolution of concentrations, and the other an
equation for the evolution of order parame-
ters. These two classes of problems, coupled
with intermediate situations, cover the range
of problems involving relaxed interfaces.
The overall method is to write a Helm-
holtz energy functional involving not only
the dependence of the local composition C
or order parameter
y, labeled f (C, y), but
also those gradient terms compatible with
the symmetry of the problem. This leads to
terms in (—C )
2
or (—y)
2
. The interfacial
energy between the two regions of different
prescribed order parameter or composition
is then computed in two steps: the concen-
tration or order parameter which minimizes
486 7 Transformations Involving Interfacial Diffusion
Figure 7-2.Top: A schematic concentration (C)
profile through a diffuse coherent interface (compo-
nents A and B) in which excess chemical Helmholtz
energy, represented by the quantity D
m, is every-
where balanced against gradient Helmholtz energy
(F: Helmholtz energy,
m
e
A
and m
e
B
: equilibrium chem-
ical potentials). Bottom: Finite thickness
dof equi-
librium interface, corresponding to concentration
profile.www.iran-mavad.com
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538. 7.2 Equilibrium Properties of Solid – Solid Interfaces 487
the excess Helmholtz energy is computed,
and this excess energy is then taken as the
interfacial energy. In a single model we
therefore get both a “continuum structure”
for the interface, and the specific interfa-
cial Helmholtz energy.
The time-dependent version of these
equations allows us to treat the motion of
the interface under a prescribed driving
force, provided that a relation is assumed
between the fluxes and the driving forces.
For a conserved order parameter C(such
as concentration), the kinetic equation
takes the form:
(7-2)
where
kis a gradient energy coefficient,
and for a non-conserved order parameter
(such as a long-range order parameter, or a
crystal orientation), we obtain:
(7-3)
The interfacial Helmholtz energy for a dif-
fuse interface between two concentrations
C
1and C
2is given by
(7-4)
Similarly, assuming a quadratic expan-
sion of the Helmholtz energy with two min-
ima for order parameters
y= 1 and y=–1
gives the following expression for the inter-
facial Helmholtz energy of an antiphase
boundary:
(7-5)
In each case, the width of the interface scales
as
k/f
1/2
: the stronger the gradient term com-
pared with the local chemical Helmholtz
energy contribution, the more diffuse will
be the interface. The “structural width” of
sk=
8
3
2
Nf
v ′′
s k=d2
1
2
2
NfCC
C
C
v
∫()
∂
∂
∇− ′y
ky y
t
Mf=[ ()]
22
∂
∂
∇−∇
C
t
MfC C=[ () ]
224
k
the interfaces is thus related to the coupling
term weighting the square gradients.
These continuum descriptions are in
principle well suited for application to dif-
fuse interfaces. Their facility of implemen-
tation in computer simulations (since they
do not involve front tracing methods) has
triggered much interest under the name of
“phase field approaches” (Carter et al.,
1997). They have recently been applied to
various problems involving sharp inter-
faces using the following systematics: the
interface is artificially made diffuse with
respect to a “phase field parameter”
j. (For
instance, this parameter can be taken equal
to 1 in a solid and to 0 in a liquid if we wish
to address a solidification problem involv-
ing both solute and heat flow (Wheeler et
al., 1992). It can be taken to vary continu-
ously from +1 to –1 through a grain bound-
ary, mimicking the orientation change
across the boundary for application to a
problem in grain growth (Chen et al.,
1998).) A Helmholtz energy equation is
proposed which involves both the concen-
tration and the phase field parameter and
their spatial derivatives. The dynamic equa-
tions for the concentrations and the phase
fields are solved using continuum methods
and the coupling term containing the phase
field gradient is set equal to zero so that the
sharp interface can be recovered.
These methods have been applied to
solidification, phase transformations, grain
growth, and chemically induced grain
boundary migration. They possess the ad-
vantage of relative ease of implementation
using classical numerical techniques, and a
great flexibility in terms of the introduction
of such features as anisotropy. They share
the drawback of all Ginzburg–Landau type
phenomenological equations, i.e., that the
precise atomistic interpretation of both the
mobility and the coupling terms is not al-
ways clear.www.iran-mavad.com
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539. When a structural difference coexists
with the chemical difference, the situation
becomes more complex. However, if the
difference between phases is small, as is
the case for structurally similar phases with
similar orientations and lattice parameters,
the differences may be accommodated by
an elastic component or by an array of mis-
fit dislocations. In the case of
gprecipitates
in
bbrass, the interfacial energy has been
taken as the sum of structural (dislocation)
and chemical energies. This approach per-
mits the prediction of weak anisotropies,
for example those in Fig. 7-3, after Ste-
phens and Purdy (1975).
More complex systems, e.g., interfaces
joining f.c.c. and b.c.c. phases, are gener-
ally discussed with respect to a particular
orientation relationship or set of orienta-
tion relations, considered to be established
at nucleation. In these cases, we search for
a reason for the strong anisotropy of inter-
facial Helmholtz energy commonly ob-
served in such systems. The most plausible
approach involves the search for optimum
matching of atomic positions at the inter-
face, that is a maximum in coherency, cor-
responding to a minimum in the structural
component of the energy. A particularly
simple example is found in the f.c.c./h.c.p.
system Al–Ag, where the (111) f.c.c. habit
plane corresponds to the structurally simi-
lar (001) h.c.p.
Figure 7-4 illustrates a model of the
f.c.c./b.c.c. interface due to Rigsbee and
Aaronson (1979), in which a macroscopi-
cally irrational habit plane interface is
made up of rational areas with a high de-
gree of coherency, separated by “structural
ledges” which serve to displace the inter-
face plane by a few atomic dimensions,
thereby to increase the degree of cohe-
rency. Aaronson and his co-workers em-
phasize that these structural ledges are con-
sidered intrinsic to the structure of the
interface, and do not provide a mechanism
for interface motion. Other studies have
utilized a more formal geometrical ap-
proach (Bollman, 1970; Dahmen, 1981;
Zhang and Purdy, 1993a, b) to seek inter-
faces of optimal matching, with similar
conclusions. Based on evidence from high
resolution electron microscopy (e.g.,
Zhang et al., 1998), it has become apparent
that the typical interface between phases of
dissimilar structure is often faceted on a
microscopic scale, and that invariant
planes or invariant lines of the transforma-
tion, when they exist, are clearly important
in the selection of precipitate habits.
488 7 Transformations Involving Interfacial Diffusion
Figure 7-3.The morphologies of equilibrated gpre-
cipitates (solid curves) formed in
bbrass, and the
corresponding Wulff plots (dashed), for the tempera-
tures indicated. The anisotropies are consistent with
a model that treats the interfacial Helmholtz energy
as a sum of structural (dislocation) and chemical
terms (Stephens and Purdy, 1975).www.iran-mavad.com
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540. 7.2 Equilibrium Properties of Solid – Solid Interfaces 489
7.2.3 Segregation to an Interface
In system of two or more components, it
is often found that the equilibrium compo-
sition at an internal surface is different
from that of the surrounding phase(s). As
before, it is necessary to distinguish be-
tween grain and interphase boundaries; for
the case of a grain boundary in a two-com-
ponent solid solution, we may consider that
the high energy defect has its own thermo-
dynamic solution properties. A constrained
equilibrium is then possible between the
boundary “phase”, which must have an ap-
proximately fixed volume, and the adjacent
crystalline phase (a). The constant volume
condition requires, for a virtual change in
mole number n
a
A
,
dn
a
A
= – dn
b
A
;dn
b
A
= – dn
b
B
;
dn
b
B
= – dn
a
B
(7-6)
and the constrained equilibrium is defined
by a line tangential to the aHelmholtz-en-
ergy curve at the bulk composition C
0, and
a parallel line tangential to the boundary
Helmholtz-energy curve (Hillert, 1975a).
This is illustrated in Fig. 7-5. We then ob-
tain for the distribution coefficient
(7-7)
s
C
C
=
B
b
B
a
(7-8)
where
s
Aand s
Bare the specific grain
boundary Helmholtz energies in pure A and B, V
b
m
the molar volume of the boun-
dary “phase”, and L
a
and L
b
the regular so-
lution parameters for the crystal and boun- dary phases, respectively.
A more comprehensive treatment of
phase equilibria of grain boundary struc- tures is given by Cahn (1982). The pos- sibility of faceting transitions is high- lighted in this work and supported experi- mentally by the studies of Ferrence and Balluffi (1988) and Hsieh and Balluffi (1989).
ln
()
s
V
RT
LL
RT
=
BAm
bbss
d−
+
−
a
Figure 7-4.Rigsbee and
Aaronson’s model for the
f.c.c./b.c.c. interface, in
which structural ledges are
introduced which maximize
the degree of coherence
within the interface. Re-
printed with permission
from Rigsbee and Aaronson
(1979), copyright Pergamon
Press.
Figure 7-5.Graphical interpretation of the con-
strained equilibrium between a crystalline solid solu- tion, a, and a grain boundary phase, b. (For details
see text.)www.iran-mavad.com
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541. For interphase boundaries, segregation
is also expected, as first shown by Gibbs
(1906). His expression for the relationship
between the rate of change of interfacial
Helmholtz energy with bulk composition
and the interface excess solute concentra-
tion,
G
B, the Gibbs absorption isotherm,
may be written as
d
s= –G
Bdm
B (7-9)
where
G
Bis the surface excess concentra-
tion of solute B per unit area of interface.
(The solvent surface excess is zero by defi-
nition.)
A growing body of experimental evidence
demonstrates that both grain boundaries
and interphase boundaries are generally
sites for complex interactions among chem-
ical species. Guttman (1995) has reviewed
the elementary mechanisms of intergranu-
lar segregation in multi-component sys-
tems, and concludes that site competition,
as well as attractive and repulsive chemical
interactions, are all important in determin-
ing the degree of interfacial segregation.
7.3 Interfacial Diffusion
Solid–solid interfaces often provide ex-
ceptionally rapid paths for diffusion. Pro-
vided that there is a significant structural
difference, so that there is a disordered re-
gion linking nearly perfect crystalline re-
gions, this is easy to understand in broad
terms. The mechanisms and the details of
grain and interphase boundary diffusion
are not fully understood at present, al-
though there exists a wealth of experimen-
tal information on the former. The reviews
by Peterson (1983) and Kaur and Gust
(1988) for grain boundary diffusion, and
that of LeClaire (1986) for the related
problem of diffusion along dislocations are
recommended.
Grain boundary diffusion is commonly
investigated experimentally in one of two
ways:
a) “Random” polycrystalline specimens
are exposed to a source of diffusant (e.g.,
a planar plated or implanted layer), an-
nealed, then serially sectioned. If the dif-
fusant is a tracer, the technique can be
used to give a sensitive evaluation of the
product D
–
b
d, where D
–
b
is the average
grain boundary diffusion coefficient, and
dthe boundary thickness, assumed to
represent a region of uniformly enhanced
diffusivity. If a solute is diffused into the
polycrystal, the usual experimentally ac-
cessible quantity is the product {sD
–
b
d}
(Levine and MacCallum, 1960). It is only
in exceptional cases, where the volume dif-
fusion contribution to some part of the pen-
etration profile is negligible (type “C” be-
havior) that we can obtain in principle a
more direct evaluation of the boundary dif-
fusivity.
b) Bicrystals of controlled misorienta-
tion are sectioned such that the grain boun-
dary plane is perpendicular to a source of
diffusant, which may be of constant com-
position, or finite. In most cases, as above,
both volume diffusion and boundary diffu-
sion will contribute to the penetration pro-
file, as schematized in Fig. 7-6. In this
more controlled case, the mathematical
solutions of Whipple (1954) or Suzuoka
(1964) (depending on whether the source is
fixed in composition or finite) are used to
unfold the grain boundary transport coeffi-
cient from serial sectioning data. The dif-
ference here lies in the opportunity to cor-
relate misorientation (and, by inference,
structure) with diffusion rates. Again, a
large amount of precise data has been ac-
quired, mainly for the special case of the
symmetric tilt boundary. This was re-
viewed by Peterson (1983).
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542. 7.3 Interfacial Diffusion 491
From the consideration of grain bound-
ary diffusion data for metals, we can draw
a few general conclusions:
a) The activation enthalpy for grain
boundary self-diffusion is of the order of
half that for volume diffusion. Indeed, a
good empirical correlation exists between
grain boundary diffusivity and reduced
temperature T/T
mfor different metal struc-
tures. For f.c.c. metals, Gust et al. (1985)
obtained
(7-10)
dD
–
b
= 9.7 ¥ 10
–15
exp(–9.07T
m/T)m
3
/s
b) For the low angle symmetric tilt
boundary, rates of diffusion are aniso-
tropic, with faster diffusion occurring par-
allel to the dislocation lines. Undissociated
dislocations have higher diffusivities than
dissociated ones; the diffusivity increases
linearly with dislocation density (and there-
fore with tilt angle
q) for angles up to
about 10°.
c) Anisotropic behavior is also obtained
for higher angle symmetric tilt boundaries.
d) Low angle twist boundaries, com-
posed of screw dislocations, are less effi-
cient in transporting material than their tilt
counterparts.
e) The mechanism of grain boundary dif-
fusion in close-packed metallic systems
is thought to involve vacancy exchange.
This is a tentative conclusion based on
the experimental evidence of Martin et al.
(1967), who studied the pressure depen-
dence of grain boundary diffusion and de-
duced activation volumes from their data,
and of Robinson and Peterson (1973), who
studied isotope effects in silver polycrys-
tals and bicrystals, and demonstrated that
their results were consistent with a vacancy
mechanism. In their computer simulation
studies, Balluffi et al. (1981) and Faridi
and Crocker (1980) have provided support
for this idea by showing that the formation
and migration energies for vacancies and
interstitials in the grain boundary favored a
vacancy exchange process.
Having summarized briefly the present
level of understanding of grain boundary
diffusion, it is perhaps worth emphasizing
that the correlation of structure with diffu-
sivity remains incomplete, and that much re-
mains to be learned about the fundamental
processes of interfacial diffusion in solids.
Interphase boundary diffusion is less
well studied, to the extent that very few
reliable data are available in the litera-
ture. Kaur and Gust (1988) cite only one
system for which extensive measurements
have been reported, the Sn–Ge/In system
(Straumal et al., 1981). One reason for the
lack of direct information about this impor-
tant aspect of interfacial diffusion may lie
in the fact that, unlike grain boundaries, in-
terphase boundaries can support local equi-
librium concentration gradients only in the
presence of gradients of curvature, temper-
ature or stress. The isothermal planar inter-
phase boundary is normally isoconcentrate.
Mullins (1957) has considered a number
of important cases of interfacial diffusion
in response to variations in curvature. For
Figure 7-6.Schematic composition (C) distribution
in the neighborhood of a grain boundary, normal to a
surface of fixed composition, C
0. d: boundary thick-
ness.www.iran-mavad.com
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543. the case of substitutional diffusion, and
under the simplifying assumptions of con-
stancy of composition of the two phases a
and band site conservation at the interface,
we may write for the interfacial flux:
(7-11)
Here, Lrepresents the mobility of atoms
along the interface, V
mis a molar volume,
the
m’s are chemical potentials within the
interface,
s
abis the specific interfacial
Helmholtz energy and K
abis the interfacial
curvature.
7.4 Forces for Interface Migration
It is useful to define and classify a set of
generalized “forces” for interface migra-
tion. In most cases, the driving and retard-
ing forces are expressed in terms of a nor-
mal force per unit area of interface, and
therefore have the units of pressure.
It is conventional to separate the various
forces according to their origin (e.g., surface
Helmholtz energy, elastic energy, chemical
Helmholtz energy) and to some extent the
division is arbitrary. Its utility lies in the
ability to visualize the interplay of different
forces on a moving transformation interface.
It is also true that the definition of the
“force” for boundary migration is only a
part of the solution to the migration prob-
lem. The response function (or mobility) of
the interface must also be known, and this
function is expected to differ greatly from
one interface type to the next.
7.4.1 Capillary Forces
These forces originate from the system’s
tendency to reduce its total interfacial
JJL
LV
cc
K
ab ab ab ab ab
ab ab
a b
ab
AB AB
m==
=
−−∇−
−
−
∇
() mm
s
Helmholtz energy sA, where sis the spe-
cific interfacial Helmholtz energy and A
the interfacial area. For constant
s, the
force can be expressed as
p
s= Ks (7-12)
which is the product of the surface Helm-
holtz energy and the interface curvature
K=dA/dV.
For smoothly curved interfaces the curva-
ture is the sum of the reciprocals of the two
principal radii of curvature [(1/r
1)+(1/r
2)].
For strongly anisotropic or faceted inter-
faces, an effective capillary force may still
be defined, through the derivative dA/dV, al-
though care must be taken in its formulation.
For the special case of an equilibrium
particle, the capillary force gives rise to
a variation of the specific Helmholtz
energy of the included phase: the resulting
relation between the zero-curvature equi-
libria (C
0) and the equilibrium composi-
tions in the presence of curvature is sum-
marized in the Gibbs–Thompson relation-
ship. For two components we obtain the
linearized expressions (Purdy, 1971)
(7-13a)
(7-13b)
where ais the included phase and bthe
external phase, V
mis the molar volume of
the aphase, V′
mthe molar volume of the
aphase extrapolated to the composition of
the bphase and , where
g
bis an activity coefficient. This last term
is a measure of the departure of the parent
phase from ideal solution behavior.
An equivalent graphical construction
due to Hillert (1975a) is shown in Fig. 7-7.
e
g
b
b
b=1+
∂
∂
⎛
⎝
⎜
⎞
⎠ ⎟
ln
lnC
D
D
a
aa
a b
b
b
bb
a b
b
C
CCV
CC
K
RT
C
CCV
CC
K
RT
=
=
m
m
00
0 0
00
0 01
1
()
()
()
()
−
−
⎛
⎝
⎜
⎞
⎠
⎟
− ′
−
⎛
⎝
⎜
⎞
⎠
⎟ s
e
s
e
492 7 Transformations Involving Interfacial Diffusionwww.iran-mavad.com
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544. 7.4 Forces for Interface Migration 493
The construction is of great conceptual
value, and permits the straightforward ex-
tension to the case of three components
with different partial molar volumes. Mor-
ral and Purdy (1995) have also developed
compact expressions for the general multi-
component case. For example, the solubil-
ity change of the included aphase (a
column vector |D C
j
b|) with curvature can
be written in terms of second derivatives
of the Gibbs energy, , of the par-
ent bphase, the molar volume of the a
phase, V
m, and the equilibrium concentra-
tion differences between the a and b
phases, the row vector |DC
j
ab|:
(7-14)
For faceted equilibrium shapes, the cur-
vature may be replaced by a constant term
s
i/r
i, where the subscript indicates facet i,
and r
iis the minimum radial distance to the
ith facet.
DD
ab b
C
G
C
CKV
j
j
k
j
∂
∂
2
2
=
ms
∂
∂
2
2
G
C
j
k
7.4.2 Chemical Forces
The chemical force for interface migra-
tion should be distinguished from the
chemical driving force for the transforma-
tion DF
ch, defined by Eq. (5-9) in the chap-
ter by Wagner et al. (2001). In this section
the case where the chemical force acts di-
rectly across the transformation interface is
considered. Its magnitude is given by
(7-15)
where DF¢
chis the molar Helmholtz energy
difference acting directly across the inter-
face, and is generally less than DF
ch, the
Helmholtz energy difference accompany-
ing the complete transformation. The force
is derived through a virtual work argument,
in which the energy change in displacing
the interface through unit normal distance
is equated to a work term.
The distinction between total and local
chemical driving forces is best seen in the
composition-invariant transformation il-
lustrated in Fig. 7-8. Here, the Helmholtz
energy difference for interface migration is
given by DF¢, and some fraction DF
0
of
the total available Helmholtz energy is dis-
sipated in the volume diffusion field. In the
often assumed limit of local equilibrium,
DF¢is implicitly set equal to zero, and all
of the Helmholtz energy difference for the
transformation is used in the volume diffu-
sion field.
7.4.3 Mechanical Forces
These forces fall into two categories:
those that may simply be treated as part of
the chemical Helmholtz energy of the
strained phase(s) and those that cannot.
7.4.3.1 Elastic Strain Energy
As noted by Wagner et al. (2001, Chap. 5
of this volume), elastic strain energy is an
p
F
V
ch
ch
m=
D′Figure 7-7.Effect of interfacial curvature, K, on the
two-phase equilibrium in a binary system; the com-
mon tangent line, which generates equilibrium com-
positions at its points of tangency for the planar
phase interface, is replaced by two nonparallel tan-
gent lines, whose intercepts differ by the amounts
shown.www.iran-mavad.com
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545. additive term in the Helmholtz energy of a
phase. Thus, for homophase variations, the
local elastic energy density is given by
h
2
DC
2
Yas indicated in Eq. (5-12).
This term generally reduces the total
driving force for a phase transformation. In
Eq. (5-12), DCis the local difference from
the average concentration C
0. The equation
therefore has application to any case where
a gradient of misfitting solute exists in an
otherwise perfect crystal, and in particular
to the case where a thin layer of altered
composition is in coherent contact with an
infinite bulk phase of composition C
0.
The strain energy density in a coherent
solute profile will be given as an intensive
variable by Eq. (5-12), which therefore has
application to the diffusional growth of a
precipitate in a binary solid solution, and
also to the problem of the stability of thin
epitaxial deposited layers.
For the more complex case of the strain
energy associated with the formation of a
second phase inclusion, Eq. (5-14) holds,
independently of inclusion shape, when the
transformation strain is a pure dilation and
the elastic constants of the two phases are
similar. For spherical coherent transformed
regions, the combined effects of surface
energy and (dilatational) transformation
strain may be incorporated in the Gibbs–
Thomson equation by adding 6H
e
2
g
Bto
the numerator of the term in braces in Eqs.
(7-13). Here H=3
k
a/(3k
a+4g
B) where
k
ais the bulk modulus of the precipitate,
g
Bis the shear modulus of the matrix, and e
is the misfit (Rottman et al., 1988).
7.4.3.2 Plastic Response
It is important to distinguish between
mechanically derived forces which origi-
nate from elastic strain energy, and those
which result in the plastic relaxation of the
loaded specimen. In the latter case, the
elastic energy may remain approximately
constant during the course of the transfor-
mation, but the interfaces can experience
virtual forces, in the sense that the Helm-
holtz energy of the loading system is re-
duced by interface motion.
As an example, consider the symmetric
tilt boundary of Fig. 7-9, which is capable
of motion in response to the applied load.
Studies of this type of boundary have
shown that the dislocation boundary under-
goes normal migration, that is, synchro-
nous glide, under an appropriate load. We
can consider that there is a virtual mechan-
494 7 Transformations Involving Interfacial Diffusion
Figure 7-8.Helmholtz energy and interfacial com-
position relationships for a composition-invariant
phase transformation. The available Helmholtz en-
ergy for the transformation is used in part to drive the
interface, DF¢, and the balance, DF
0
, is dissipated in
the volume-diffusion field.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

546. 7.4 Forces for Interface Migration 495
ical force on the interface equal to p
w=W
tan
qwhere Wis the applied load per unit
area and
qis the angle of tilt. In this exam-
ple, the applied stress can be thought of
as acting directly on the interface. The ex-
perimental response of the symmetric tilt
boundary is linear, and a simple statement
of the mobility of the boundary summa-
rizes the kinetics of boundary motion.
If a coherent twin boundary were to re-
place the symmetric tilt boundary in Fig.
7-9, the transformation strain would again
be a shear. It is clear, however, that the syn-
chronous glide process would not occur.
The propagation of the twin will require
the lateral motion of twinning dislocations,
and the interface response function will be
fundamentally different. This simple ex-
ample illustrates the general principle that
a knowledge of the force for boundary mi-
gration is only part of the empirical de-
scription of boundary motion; evidently the
response characteristics of the boundary
must also be known.
Another class of strain energy stored in
bulk material is that associated with struc-
tural defects such as dislocations. These
defects may be generated and stored during
plastic deformation, and their effect on the
Helmholtz energy of the solid provides the
basis for the driving force for recrystalliza-
tion. Dislocations may also occur due to
plastic relaxation of misfit stresses (as seen
in some cases of discontinuous precipita-
tion or in some transformations in steels).
The excess Helmholtz energy is tradi-
tionally taken to be proportional to the total
dislocation density and is added as a bulk
driving force in the overall thermodynamic
balance. This is probably acceptable for
misfit-generated dislocations but, for the
case of dislocations stored during plastic
deformation, this approximation neglects
the formation of dislocation substructures
such as subgrains. This relaxation phenom-
enon decreases the stored Helmholtz en-
ergy and renders its increase with disloca-
tion density less than linear due to self-
screening effects (Verdier et al., 1997).
Again the question of the contribution of
this stored energy to the overall kinetics of
microstructure development is two-fold:
we may consider the nucleation step from
an initially homogeneous situation, or the
growth regime for which the stored energy
is different on either side of the moving
interface. A simple approximation for the
growth process would be the addition of an
extra driving force when the stored ener-
gies differ on either side of the interface
(e.g. due to different stages of deformation
in two neighboring grains). However, this
approach is unlikely to be sufficient to ex-
plain the nucleation of recrystallization,
which is known to occur at grain boundar-
ies, and at localized shear or transition
bands (Humphreys and Hatherly, 1995). As
proposed initially by Bailey and Hirsch
(1962), nucleation seems to be a random
Figure 7-9.A tilt wall (tilt angle q) is subject to a
virtual force, p, derived from the work done on the
specimen by the loading system.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

547. event associated with local fluctuations in
subgrain configuration; it is unlikely to be
successfully modeled either by a standard
nucleation theory or by an averaging ap-
proach for the stored energies. Information
on local misorientations (leading to differ-
ences in local mobilities and local driving
forces) may be crucial, as has been sug-
gested using vertex-based numerical simu-
lations (Weygand, 1998).
7.4.4 Frictional Forces
7.4.4.1 Solute Drag
Each type of mobile interface will pos-
sess its own characteristic response func-
tion. For the simple tilt wall subject to a
mechanical force, we have noted that the
force–velocity relationship is linear over a
range of forces. If a considerable recon-
struction of the crystal structure is required
for the motion of an interface, as in the mo-
tion of a high-angle boundary in a single-
component system, we expect that cross-
boundary diffusive motion will play a sig-
nificant role. Turnbull’s (1951) expression
for the intrinsic mobility, M , of a grain
boundary
(7-16)
is based on a model in which atom jumps
across the boundary are independent of one
another. This relation has not been con-
firmed by experiment, and this is taken to
suggest that more complex, cooperative
movements of atoms are generally in-
volved in interface migration. Again refer-
ring to experiment, there exists a consider-
able database drawn from measurements
of grain boundary motion in high purity
metals. Thus the intrinsic response of
high-angle grain boundaries is rather well
documented, even if the mechanisms of
M
DV
bRT
=
b
m
d
2
boundary migration are not fully under- stood.
When two or more components are in-
volved, this picture must be extended. Superimposed on the intrinsic response of the interface, and frequently masking it, we find the effects of solutes which are prefe- rentially attracted to or repelled from the interface. The experiments of Aust and Rutter (1959) first demonstrated conclu- sively the powerful retarding effects of trace amounts of solute on grain boundary migration.
The idea of a “solute drag” perhaps orig-
inated with Cottrell (1953), who noted that solute atoms would be attracted to disloca- tions, and would therefore be required to diffuse along with the dislocations, or to be left behind by dislocations that had broken free of their solute atmospheres. Similar ideas were put forward by Lücke and his co-workers (1957) for application to the case where a solute is preferentially segre- gated to grain boundaries. The most com- plete and informative theoretical treat- ments of this effect are those of Cahn (1962) and Hillert and Sundman (1976). The latter is based on a treatment put for- ward earlier by Hillert (1969). The basic reason for the solute drag force is simple: if solute is attracted to the boundary, it will tend to diffuse along with the moving boundary. Depending on the relative rates of boundary motion and solute diffusion, the solute distribution may become asym- metric, with more solute trailing the boundary than leading it. Because of the mutual attraction between the boundary and the solute, the asymmetric solute dis- tribution leads to a retarding force of the form
p
i= N
VÚ(C–C
0)(dE/dx)dx (7-17)
where N
Vis the number of atoms per unit
volume, and dE/dxis a measure of the
496 7 Transformations Involving Interfacial Diffusionwww.iran-mavad.com
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548. 7.4 Forces for Interface Migration 497
force between the solute atoms and the
boundary. The effects of small amounts of
solute can be profound. The profiles in Fig.
7-10 are due to Cahn (1962). They are
based on a triangular interaction potential
between solute and boundary, and corre-
spond (a) to a slowly moving boundary, (b)
to a boundary experiencing maximum so-
lute drag, and (c) to a fast-moving bound-
ary. That is, the drag force goes through a
maximum as the velocity is increased. If an
initial intrinsic force–velocity relationship
is assumed linear, and the solute drag term
is added, the total response function is
quickly rendered nonlinear. Under more
severe conditions, the system can even be-
come unstable, displaying a region of ve-
locities where is negative.
Hillert’s formulation uses a different
interaction potential, initially square
(rather than triangular), and later a square-
topped potential with ramps at the edges.
The drag is evaluated as a rate of dissipa-
tion of Helmholtz energy due to diffusion
in and in front of the moving interface. The
model lends itself to numerical analysis; it
∂
∂
p
v
is not restricted to dilute solutions, nor is it
restricted to grain boundaries. Figure 7-11
uses the simple square well to illustrate the
effect of cross-boundary diffusion in set-
ting the degree of asymmetry of the solute
profile. The drag term may be written in
several ways. Among the most convenient,
Hillert and Sundman (1976) find:
(7-18)
In their approach, the ramps (zones 1 and 3
in Fig. 7-12), as well as the centre of the
boundary (zone 2) and the parent phase re-
gion (zone 4), contribute to the drag. In-
deed, it is possible to separate out the con-
p
RT
V
cc
Dc c
x
i=d
m
v
∫
−
−
()
()
0
2
1
Figure 7-10.Computed solute profile in the vicinity
of a moving grain boundary: a) at equilibrium, b) at a
velocity corresponding to maximum solute drag, and
c) at higher velocity. After Cahn (1962).
Figure 7-11.A square-well model for the moving
grain boundary. After Hillert (1969).www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

549. tribution of any slice taken parallel to the
interface, and Sundman and Hillert have
presented their results in this way, as com-
ponents of the total solute drag. They find,
as we might expect, that the drag is reduced
for given velocity by increasing the cross-
boundary diffusion coefficient, or by re-
ducing the depth of the potential well seen
by the solute. The model is capable of gen-
erating a variety of informative results. It
is not restricted to grain boundary motion,
but also finds application to steady state
phase transformations, as will be seen later.
It is interesting to note that solute drag
effects have been (in some sense) better
studied and quantified than intrinsic bound-
ary friction forces. The solute effects are
thought to be additive, and to dominate and
even to mask the intrinsic properties of
grain boundaries for many cases of practi-
cal interest.
7.4.4.2 Particle Pinning
Whereas the contribution of solute drag
as a retarding force is by definition of a vis-
cous nature, another term may prevent the
interface from moving: a pinning force.
The idea of pinning a grain boundary with
foreign particles was put forward initially
by Zener (Smith, 1948). The grain bound-
ary will be able to move only if the gradient
of bulk stored energy across the interface
is sufficient to move the interface to an
untrapped configuration. Zener assumed a
purely geometrical distribution of particles
along the grain boundary; this led to a fric-
tion force to be overcome that scales as
V
f/r, where V
fis the volume fraction of
particles and rtheir radius. A more refined
approach due to Hazzledine and Oldershaw
(1990) allows us to treat the collective pin-
ning problem in a consistent manner (i.e.,
to treat both the flexibility of the interface
and the number of particles along the inter-
face). Hazzledine’s result, recently con-
firmed by computer simulations (Weygand,
1998), indicates that a scaling law in V
f
1/2/r
might be more appropriate.
The consequence of these pinning forces
on the overall kinetics of interface migra-
tion can be two-fold: on the one hand, the
particles will impose a threshold driving
force for boundary migration; and on the
other hand, they will impose a constant re-
tarding force to be subtracted from the
available Helmholtz energy. The standard
method is to equate the retarding force with
the threshold force, but this identification
is by no means obvious. Numerical experi-
ments on grain growth kinetics do however
indicate that this simple approach gives a
good description of the overall kinetic be-
havior (Weygand, 1998).
7.5 Examples
7.5.1 The Massive Transformation
Massive transformations are considered
to include all inhomogeneous, noncoher-
ent, thermally activated, composition-in-
498 7 Transformations Involving Interfacial Diffusion
Figure 7-12.The Hillert–Sundman (1976) model of
a moving boundary, showing the interaction profile,
and some computed solute profiles for v
d/2D=0.001
(1), 1.61 (2), and 10 (3).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

550. 7.5 Examples 499
variant, solid–solid transformations (Mas-
salski, 1958). Massive solidification reac-
tions are excluded by this definition, but
their literature is relevant, in part because
of advances in understanding of the rapid
solidification of silicon-based alloys (Aziz
and Kaplan, 1988).
The massive transformation generally in-
volves a major structural change; it normal-
ly begins with the heterogeneous nuclea-
tion of a thermodynamically stable or meta-
stable daughter phase, at imperfections or
grain boundaries in a supercooled parent, and
proceeds by the thermally activated migra-
tion of a mobile transformation interface.
Plichta et al. (1984) reviewed the avail-
able information on massive transforma-
tion nucleation, and concluded that the
nucleation event is structurally identical
with that expected for diffusional transfor-
mations among dissimilar phases; grain
boundary and triple junction sites are ener-
getically preferred, and an orientation rela-
tionship between the nucleus and at least
one parent grain is generally set on nuclea-
tion. Since interfacial torques are likely to
be present during the nucleation event, the
construction of Hoffman and Cahn (1972)
may be expected to provide guidance in the
modeling of the critical nucleus. Experi-
ence has shown, however, that the search
for plausible nucleus shapes in such heter-
ogeneous systems seldom leads to unam-
biguous results.
The chemical Helmholtz energy change,
DF
ch, for nucleation in a single-component
system that undergoes an allotropic trans-
formation is obtained directly from the
undercooling below the equilibrium tem-
perature, as suggested by Fig. 7-13. For
small undercoolings, DT, this may be ex-
pressed as
DF
ch= DHDT/T
0 (7-19)
with D Has the change of enthalpy.
For two components, the composition-
invariant condition need not apply to mas-
sive nucleation (although it must hold
overall for massive growth). In the exam-
ple in Fig. 7-14, drawn for a temperature
below T
0, the temperature for which the a
and bphases of the bulk composition C
0
have the same molar Helmholtz energy, it
is clear that the formation of a nucleus of
composition C
0would be accompanied by
a volume Helmholtz energy change
DF
1/V
m. However, the volume Helmholtz
energy change is maximized (DF
2/V
m) if
the nucleus takes composition C¢. Even in
the single-phase supersaturated region, a
similar argument shows that the volume
Helmholtz energy change will be max-
imized for a composition other than C
0.
Turning to the process of massive prod-
uct growth, if we again consider the allo-
tropic transformation of pure element, it is
clear that the rate of transformation is de-
termined by the undercooling (which sets
the interfacial driving force p
ch), and by the
migration characteristics of the transforma-
Figure 7-13.Helmholtz energy relationships for the
allotropic transformation of a pure element.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

551. tion interface. As in the case of the high an-
gle grain boundary, the details of the cross-
boundary atomic transfer process are not
well understood. As in the grain boundary
case, it has proved difficult to separate the
effects of trace impurities from the intrinsic
migration properties of the noncoherent
interphase boundary. Hillert (1975b) has
estimated the mobility of grain boundaries
in pure iron from a series of limiting argu-
ments as
M= 0.035 exp (–17 700/T)m
4
/(J s) (7-20)
He then used this expression in his anal-
ysis of the rates of the a/gtransformation
of iron with very low carbon contents, as
investigated experimentally by Bibby and
Parr (1964) and Ackert and Parr (1971).
Using a model in which a fraction of the
chemical force is dissipated by solute dif-
fusion ahead of the interface, Hillert was
able to predict a number of characteristics
of the massive transformation in pure and
nearly pure iron, as summarized in Fig. 7-15.
For richer alloys, massive growth could,
in principle, be entirely composition-invar-
iant. However, some results (Singh et al.,
1985) suggest that solute diffusion ahead
of the interface can play a major role in
steady interface migration. Referring to
Fig. 7-16, which is based on the simple
model of an infinitely thin interface, it is
clear that a solute build-up in front of the
interface will result in a reduction of the
chemical force on the interface. Having
conceptually released the concentration in
the interface region from its value far from
the interface (C
0), the description of two-
component massive growth becomes a free
boundary problem, in which the interfacial
driving force and the velocity must be si-
multaneously determined. It is likely that
this type of approach to local equilibrium
occurs at higher relative temperatures,
where the possibility of volume-diffusion
loss is strongest, and where, for example,
Widmanstätten growth competes with the
massive reaction.
Much of our current knowledge of mas-
sive growth derives from the pioneering
work of Massalski and co-workers (Mas-
salski, 1958; Barrett and Massalski, 1966).
500 7 Transformations Involving Interfacial Diffusion
Figure 7-14.Helmholtz energy relationship for the
isothermal nucleation of a massive product in a bi-
nary system.
Figure 7-15.Diffusional and diffusionless growth
of afrom giron, calculated assuming that carbon dif-
fusion is negligible at high growth rates. Reprinted with permission from Hillert (1975b), p. 12.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

552. 7.5 Examples 501
In early papers, it was implicitly assumed
that a pseudo-unary cross-boundary diffu-
sion process controlled the rate of massive
growth; later (Massalski, 1984) it was
noted that local changes of composition in
the region of the interface, consistent with
overall composition invariance, are pos-
sible, even probable, under certain condi-
tions.
Because of the speed of transformation,
massive reactions are often studied under
conditions of continuous cooling. The most
informative kinetic studies, however, are
those that yield rates of isothermal trans-
formation, as exemplified by the work of
Karlyn et al. (1969); they used pulse-heat-
ing and rapid quenching to evaluate the
rates of formation of massive ain a Cu–
38 at.% Zn alloy, which had previously
been quenched to retain the bphase. They
found:
i) that the transformation occurred only
within the single phase aregion of the
phase diagram and
ii) that steady massive growth (at con-
stant isothermal velocity of order 1 cm/s)
developed after a delay time of several mil-
liseconds.
Both the limited temperature range for
growth and the delay time were attributed
to the pre-existence of solute fields around
small aparticles. Only in the single-phase
aregion would such solute fields be con-
sumed during initial growth, thereby per-
mitting the development of composition-
invariant (massive) growth. They noted
that massive growth in the two-phase a+b
region (below T
0) is possible, provided that
some massive ahas first been formed by
pulsing into the single-phase region. With
further heating these massive regions con-
tinued to growth into the a+bregion, thus
reinforcing the idea that the inhibition of
such growth lies in the initiation stage.
In the analysis of growth rates, it was as-
sumed that massive growth occurred with-
out local composition change. However, in
discussion, impurity drag effects and par-
ent-phase solute fields were admitted as
possibilities.
A comprehensive theoretical treatment
of the two-component massive transforma-
tion, which applies to a range of transfor-
mation conditions, will deal with solute
diffusion within the interface and also in
the thin region ahead of it. The model of
Hillert and Sundman (1976), an extension
of their treatment of solute drag at grain
boundaries, finds application here. Their
formulation of the solute drag involves the
computation of the dissipation of Helm-
holtz energy due to diffusion within the
Figure 7-16.Solute build-up in front of the massive
interface will result in a reduction of the chemical
force for interface migration.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

553. interface region (Eq. (7-18)). The solute
field is schematized in Fig. 7-17(a) for a
solute that is attracted to the interphase
boundary. The cross-boundary diffusion of
solute introduces a component of drag (re-
gions 1, 2 and 3), and diffusion ahead of
the boundary provides a further contribu-
tion (region 4), which dominates the total
drag as the velocity approaches zero. The
total drag is optimized for and it
may be that the optimum of dissipation of
vd
2
1
D
≈,
Helmholtz energy defines the most prob-
able velocity. A second treatment of the so-
lute drag in massive growth (Bréchet and
Purdy, 1992) extended Cahn’s (1962) anal-
ysis, and demonstrated that a finite drag
term will always be present, however small
the velocity. This analysis, like that of Hil-
lert and Sundman, therefore suggests that a
threshold driving force is a natural charac-
teristic of such transformations. However,
if the solute field in front of the interface is
accompanied by a misfit, in the sense of
Eq. (5-12), we expect an additional term in
the force balance, a “pulling” force due to
the elastic energy contained in the coherent
composition gradient.
The above discussion is based on the
premise that the migration characteristics
of the massive front are those of a non-co-
herent interface. This is borne out by nu-
merous observations (Massalski, 1984)
which indicate, for example, that the trans-
formation interface is able to cross grain
boundaries in the parent phase without a
change in velocity or morphology. Dy-
namic observations indicate, however, that
the motion of the front is irregular, and of-
ten accomplished by a lateral process, in
which diffuse steps move parallel to the
interface plane (Perepezko, 1984). In the
view of Menon et al. (1988), lateral pro-
cesses are the rule in massive growth.
Hence models based solely on diffusion
normal to the boundary may need to be
modified. However, it is interesting that
Perepezko (1984) demonstrated a scaling
relationship for a wide range of alloy
systems, which yields a composite en-
thalpy of activation for massive propaga-
tion of 94T
mJ/mol. This value is similar to
that for grain boundary diffusion, which
suggests that diffusion within the interface
is a common rate-determining feature for
all such transformations.
502 7 Transformations Involving Interfacial Diffusion
Figure 7-17.a) Schematic concentration profile
through a massive transformation interface, i) at
equilibrium, and ii) in motion. b) Computed solute
drag for the massive transformation in part a, for a
constant value of D. X
·
Aand X
·
Brepresent the initial
composition of the parent phase expressed as mole
fractions of the two components A and B. After Hil-
lert and Sundman (1976).www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

554. 7.5 Examples 503
7.5.2 Chemically-Induced Grain
Boundary Migration
Chemically-induced grain boundary mi-
gration (CIGM), or diffusion-induced grain
boundary migration, is a rather recently
recognized phenomenon (den Broeder,
1972; Hillert and Purdy, 1978). The pro-
cess is one in which diffusion along a grain
boundary, to or from a sink or source of so-
lute, causes the boundary to move. The
source may be in the solid, liquid or gas
state. A composition change results (Fig.
7-18).
The process is widespread in binary me-
tallic systems. King (1987) lists 30 systems
in which it has been detected. The process
is capable of generating substantial inter-
mixing where little or none might be ex-
pected in the absence of grain boundary
motion.
A variation on the conditions in Fig.
7-18 is obtained for a supersaturated solid
solution: an initially planar boundary bows
out between grain boundary precipitates,
and simultaneously sweeps solute to the
precipitates (Fig. 7-19). This type of mi-
crostructural development was reported by
Hillert and Lagneborg (1971), and was re-
cently supported by microanalytical evi-
dence for solute depletion in the volume
swept by the moving boundary by Solor-
zano et al. (1984).
CIGM is often seen as symmetry break-
ing. The initiation process is not well docu-
mented or understood. It is clear that diffu-
sion along an initially stationary boundary
will lead to symmetric diffusion profiles,
as illustrated in Fig. 7-6, and this has led to
the suggestion that the region next to the
boundary will become one of increased
elastic strain energy, provided that a solute
misfit exists, and provided that the solute-
enriched or solute-depleted regions remain
coherently connected with their respective
grains. The strain energy density will then
be given by Eq. (5-12). Because the elastic
constant Yis a function of orientation, the
strain energy density will in general be dif-
ferent on the two sides of the boundary, and
this may lead to boundary displacement.
Tashiro and Purdy (1987), in a survey of
binary metallic systems, found only one
system in which CIGM could not be in-
duced; this was the system with the lowest
misfit, Ag–Mn.
Further evidence for the importance of
solute misfit in the initiation of CIGM is
found in the work of Rhee and Yoon
Figure 7-18.Schematic representation of chemi-
cally-induced grain boundary migration in a thin
sample exposed to a vapor source of solute. The front
is often observed to bow against its curvature.
Figure 7-19.Grain boundary bowing between pre-
cipitates, which act as solute sinks, and serve to pin the grain boundary.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

555. (1989), who systematically varied the mis-
fit parameter in a ternary system, and
showed that the phenomenon is suppressed
when the misfit is brought to zero. An ap-
proximate parabolic dependence of veloc-
ity on misfit was also found, consistent
with Eq. (5-12). This suggests that elastic
strain energy also plays a role in propaga-
tion of the grain boundary.
In practice it is difficult to distinguish
the possible driving forces for the process.
At the highest temperatures, it is likely that
solute field stresses will be dominant. This
appears to be the case for the study of Rhee
and Yoon (1989). In the early work of Hil-
lert and Purdy (1978), in which thin poly-
crystalline iron films were exposed to zinc
vapor, it was evident that the volume diffu-
sion penetration of the parent grains, in-
dexed by D
V/v, was of atomic dimensions.
Hence, it was assumed that the driving
force was entirely chemical. A comprehen-
sive model should take into account all
possible sources of the driving force.
The moving grain boundary is subject to
a set of forces, which may in general in-
clude chemical, elastic, frictional and cap-
illary forces. As in the massive transforma-
tion, there exists in principle a degree of
freedom corresponding to the concentra-
tion in the parent phase immediately adja-
cent to the boundary. In the conceptual
limit where no volume-diffusion penetra-
tion exists in advance of the boundary, the
concentration profile will be a step, and the
full chemical force p
chwill act across the
interface. In a second limiting case, a con-
centration gradient exists in front of the
boundary and an elastically-derived resid-
ual force is determined by the coherency
field. Intermediate cases would then corre-
spond to a higher or lower degree of relax-
ation of the concentration at the leading
edge of the boundary towards a constrained
equilibrium between strained parent crystal
and unstrained product crystal. These con-
siderations are contained in the phenomen-
ological treatment due to Bréchet and
Purdy (1992), in which the driving force is
evaluated over a correlation length (l) on
either side of the boundary. The steady
concentration profile is C(z). The force
then becomes:
(7-21)
where the second derivative is a measure of
the Helmholtz energy composition relation.
This approach has the virtue of including
both possible contributions to the driving
force; it results in the conclusion that, as in
the massive transformation, a threshold ex-
ists, below which no motion is possible.
An ambitious treatment of the problem
has been put forward by Cahn et al. (1998).
Their analysis is based on a phase-field
treatment of the grain boundary. They look
for travelling-wave solutions of the equa-
tions of motion, and treat their existence
as a requirement for forces capable of
coupling with the boundary motion. They
find that coupling with the elastic field is
indeed possible. However, their treatment
gives no indication that a purely chemical
force is effective in moving the boundary.
7.5.3 Discontinuous Precipitation
Discontinuous precipitation, like chemi-
cally-induced grain boundary migration,
involves the lateral diffusion of solute
within a sweeping grain boundary. The dif-
ference lies in the nature and spacing of the
solute sources/sinks. In the case of discon-
tinuous precipitation, these are members of
a regular array of precipitates, whose spac-
ing is a free variable, capable of internal
adjustment. The reaction is found in a wide
variety of precipitation systems, often at
11
2
0
2
2 0
222
l
F
C
Cz C Y Cz z
l
∫
∂
∂
−+
⎡
⎣
⎢
⎤
⎦
⎥
(() ) ()
h d
504 7 Transformations Involving Interfacial Diffusionwww.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

556. 7.5 Examples 505
low homologous temperatures where grain
boundary diffusion is expected to be a
dominant mechanism of material transport.
Like CIGM, discontinuous precipitation
can in principle occur in the absence of
volume diffusion. The reaction is capable
of destabilizing microstructure at relatively
low temperatures, and it is therefore of
practical interest.
Unlike eutectoid reactions, which yield
morphologically similar products, discon-
tinuous precipitation reactions are not as-
sociated with a particular feature of the
phase diagram, although supersaturation is
an obvious prerequisite. The question of
which systems will give rise to discontinu-
ous precipitation, and which will not, will
perhaps be answered by reference to and
detailed understanding of the initiation and
growth processes.
7.5.3.1 Initiation of Discontinuous
Precipitation
A nucleation event, in the classical
sense, is not required. As in CIGM, a pre-
existing grain boundary is caused to move,
and eventually to become the steady reac-
tion front, as suggested by Fig. 7-20. The
mechanisms proposed for the initial stages
of grain boundary displacement fall into
two broad classes: free-boundary mecha-
nisms and precipitate-assisted mecha-
nisms. Baumann et al. (1981) have indi-
cated that free-boundary initiation is domi-
nant at higher homologous temperatures,
leading eventually to “single-seam” mor-
phologies. At lower temperatures, it is ex-
pected that boundary motion is initiated by
precipitate–boundary interactions, and that
these lead to a preponderance of double-
seam morphologies. The correlation be-
tween temperature and morphology ap-
pears to be quite general, with the break oc-
curring at half the absolute melting temper-
ature. Gust (1984) found a correlation with
the solvus temperature rather than the melting
temperature. Duly and Bréchet (1994) de-
termined that the proportion of double-seam
morphologies decreased from more than
1/2 to 0 as the temperature was increased.
In free-boundary initiation, the initial
grain boundary displacement is thought to
be caused by ordinary recrystallization or
grain growth forces (Fournelle and Clark,
1972), or by forces derived from solute
segregation (Meyrick, 1976). The subse-
quent stages must then include the nuclea-
tion of precipitates at the moving boundary
and the evolution of a steady state in which
the precipitates acquire a uniform spacing,
as depicted in Fig. 7-21. Cu–Co alloys
seem to require this type of initiation, and
readily undergo discontinuous precipita-
tion only when cold-worked or when
treated to give a small parent grain size
(Perovic and Purdy, 1981).
Precipitate–boundary interactions were
first studied in detail by Tu and Turnbull
(1969), who showed that boundary torques,
Figure 7-20.Scanning electron micrograph of a dis-
continuous precipitation colony formed in Al–22%
Zn at 478 K. The original grain boundary positions is
indicated by arrows. After Solorzano et al. (1984).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

557. developed when a regular array of precipi-
tates, formed on the boundaries in Pb–Sn
bicrystals, were responsible for the initial
displacement of the boundary (Fig. 7-22).
Eventually, boundary breakaway was en-
visaged, with the subsequent embedding of
the precipitates in the advancing grain. Aa-
ronson and Aaron (1972) extended these
ideas to include a range of possible geome-
tries, each of which resulted in an initial
boundary displacement due to capillary
forces associated with the formation of
equilibrium precipitate nuclei at the grain
boundaries.
A further class of precipitate–boundary
interactions, based on CIGM, was pro-
posed by Purdy and Lange (1984). In this
case, initial displacement can occur against
capillary forces, in response to chemical
forces. If the initial precipitates are closely
spaced, the initial displacement may be
preceded by a period of precipitate coars-
ening. The process is suggested by Fig.
7-19. Several reports of grain boundaries
bowing between fixed precipitates can be
found in the literature (Hillert and Lagne-
borg, 1971; Solorzano et al., 1984). Mi-
chael and Williams (1986) found that so-
506 7 Transformations Involving Interfacial Diffusion
Figure 7-21.Free-boundary initiation
and early development of discontinuous
precipitation. Reprinted with permission
from Fournelle and Clark (1972), p. 2762.
Figure 7-22.Formation of grain boundary precipi-
tates at an initially static boundary (a), and boundary displacements in response to capillary forces (b). Re- printed with permission from Tu (1972), p. 2773.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

558. 7.5 Examples 507
lute-depleted volumes were left in the
wake of bulging grain boundaries in super-
saturated Al–4.7 wt.% Cu. Fonda et al.
(1998) and Mangan and Shiflet (1997)
studied the initiation of discontinuous pre-
cipitation in Cu–3% Ti alloys, and demon-
strated that, at low supercooling, the initial
boundary motion is due to the presence of
Widmanstätten precipitates, which grow
into one grain, and cause boundary dis-
placement into the other. Duly and Bréchet
(1994) examined the initiation of discon-
tinuous precipitation over a wide range of
temperatures, initial grain sizes and com-
positions in Mg–Al alloys. They conclude
that the Tu–Turnbull and Fournelle–Clark
mechanisms dominate at low and high tem-
peratures respectively, and that the Purdy–
Lange mechanism may be important at
intermediate temperatures.
It is now clear that a range of mecha-
nisms exist, each mechanism capable of in-
itiating boundary motion. At lower temper-
atures, precipitation on the static boundary
is a common precursor of discontinuous
precipitation; the initial precipitates are ca-
pable of acting either directly, to pull the
boundary from its initial location, or indi-
rectly through CIGM to accomplish a simi-
lar result. We conclude that, for most situa-
tions where the supersaturation is signifi-
cant, there will be no difficulty in initiating
grain boundary motion.
7.5.3.2 Theories of Steady Cooperative
Growth
The development of a steady or near-
steady growth front, characterized by a
regular spacing between precipitate lamel-
lae (or rods), S , and a constant velocity, v,
is not immediate upon initiation, but is pre-
ceded by extended transient growth re-
gions. However, steady or near-steady con-
ditions appear eventually to prevail, and
the steady state has attracted the attention
of successive generations of theorists. In
this section, we review only the more ad-
vanced theories, while acknowledging
their geneology.
Before proceeding, the question of the
steady state should be pursued in more de-
tail. It appears that the steady state can be
achieved fairly generally, and that it can be
approached from higher or lower supersat-
uration, such that the system has little or no
memory, and the steady state is characteris-
tic of the isothermal reaction conditions
only. Nevertheless, the constancy of reac-
tion front velocity has recently been ques-
tioned by a number of authors, including
Kaur and Gust (1988), Mangan and Shiflet
(1997), and Fonda et al. (1998). This point
is discussed in Sec. 7.5.3.3.
It is convenient to divide the theoretical
treatments into two classes depending on
whether the details of the transformation
front are predicted as part of the develop-
ment. In the “global” approach of Cahn
(1959), which builds on the ideas implicit
in Turnbull’s (1951) and Zener’s (1946)
treatments, the reaction front is treated as a
planar high-diffusivity path whose mobil-
ity is rate determining. Thus the important
input quantities to the theory include the
overall Helmholtz energy change accom-
panying the passage of the front and the
interface mobility. Cahn (1959) first em-
phasized the importance of storage of
Helmholtz energy in the product phase, and
showed how to estimate it from a knowl-
edge of the interphase boundary energy
s
a,b, the grain boundary diffusivity and
distribution coefficient, s , and the interla-
mellar spacing Sand velocity v. The Helm-
holtz energy balance is written
DF≤= DF
ch– 2s
abV
m/S (7-22)
Here DF
chis the total chemical Helm-
holtz energy difference between the parentwww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

559. and product phases, evaluated some dis-
tance from the interface. In determining
this quantity, the stored Helmholtz energy
in the product aphase must be calculated
from a knowledge of the composition pro-
file in the alamellae. This is determined
in theory from a knowledge of the grain
boundary transport properties and the ve-
locity and spacing through
(7-23)
where C
0is the initial concentration in the
parent phase, C(z) is the local concentra-
tion in the aphase behind the transforma-
tion front, C
3is the concentration in the a
phase in contact with the blamellae and
(7-24)
Eq. (7-23) is a solution of the moving-
boundary diffusion equation, for constant
v, S, and boundary concentration C
3; the
solution reduces in approximation to a sine
function when the centerline composition
C(z= 0) is closer to C
3than to C
0.
Cahn (1959) next chose C
3=C
e
ab
, the
equilibrium composition of ain contact
with bat a planar interface. He was then
able to evaluate the stored chemical Helm-
holtz energy in the product phases from Eq.
(7-23). To complete the description, he as-
sumed a relationship between the total
Helmholtz energy change DF≤and the ve-
locity of the form
v= M≤DF≤ (7-25)
Here, M≤is a “global” mobility that dif-
fers from the intrinsic grain-boundary mo-
bility, e.g., of Eq. (7-16).
The theory described above has been
modified by others, for example by Aaron-
son and Liu (1968), to take partial account
of capillary forces acting on the blamellae.
a
S
sD
=
b
v
2
d
CCz
CC
zaS
a
0
03
2
−
−
( ) cosh ( / )
cosh ( / )
=
a
Certain aspects of the theory are amenable to experimental verification, for example, the concentration profiles left in the a
product lamellae have been measured by Porter and Edington (1977) and by Solor- zano and Purdy (1984) using high resolu- tion elemental analysis, to give the trans- port properties of the reaction front, {sD
b
d}, through Eq. (7-23). These same
measurements allow the evaluation of the amount of the total available Helmholtz energy for the transformation retained as segregation in the product; this latter quan- tity is also available, albeit in averaged form, through X-ray measurements of the lattice parameter of the product phase.
A heuristic description of discontinuous
precipitation, after Petermann and Horn- bogen (1968), utilizes a rate equation of the type of Eq. (7-25) coupled with an approx- imate evaluation of the relaxation time for grain boundary diffusion to the blamellae,
to give a velocity expression which is di- mensionally similar to that of Cahn.
A second class of theoretical description
has been developed by Hillert (1969, 1972, 1982). This approach has been termed “de- tailed”, in the sense that a much deeper knowledge of the transformation front is implicit, even required, for its application. The basis for this treatment is the applica- tion of a local force balance at every point along the interface. Thus the possibility is raised of determining the interface shape for steady growth, as a consequence of the interplay of capillary, chemical, elastic, and frictional forces, as defined in Sec. 7.4.
In simplified form, for application to the
case where the composition profile (step), DC, measured normal to the interface is not
relaxed by volume diffusion, and where elastic and solute drag effects are negli- gible, the interfacial force balance becomes
v/M= p
ch– p
s (7-26)
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560. 7.5 Examples 509
which holds at every point. The solute pro-
file in the boundary will be given by Eq.
(7-23), and the concentration step (DC ) at
the interface will take a minimum value at
the centres of the lamellae, varying contin-
uously to a maximum value at the a/a
0
interface immediately adjacent to the bla-
mellae. Hillert has chosen the limiting
value of the solute concentration at this
point, C
3in Eq. (7-23), as the capillary-
modified solubility in ain equilibrium
with b under curvature 2/S
b.
For a steady transformation, the right-
hand side of Eq. (7-26) is constant along
the interface, and the curvature must vary
to match the chemical force, indexed by the
composition step DC.
A further local force balance is required
at the a
0/a/bjunction. This is depicted
by a vectorial balance of surface energies
in Fig. 7-23. Thus the angles of intersec-
tion and local curvatures are set at the junc-
tion, and the variation of the grain bound-
ary curvature is required to match the vari-
ation of chemical force along the boundary.
The theory therefore allows the self-con-
sistent calculation of the interface shape
for given values of the spacing S, the veloc-
ity v, and the interfacial transport property
sD
b
d.
Sundquist (1973) extended Hillert’s
treatment to include the possibility of
solute drag forces and severely nonplanar
shapes, thus allowing for consideration of
morphological stability of the transforma-
tion front.
Neither of the approaches described
above is sufficient to predict the transfor-
mation state that will occur at given super-
saturation in a particular system. Each is
capable of generating a unique relationship
between vand S, however.
In considering the spacing problem for
lamellar eutectoids, Zener (1964) set a
thermodynamic limit, a minimum spacing
for which all of the available Helmholtz
energy for the transformation is stored as
interfacial Helmholtz energy in the prod-
uct. This minimum also exists for discon-
tinuous products:
(7-27)
At this (virtual) state, the system is in
equilibrium and no growth is possible.
Larger spacings corresponds to finite
rates of growth. It has been suggested that
the steady spacing is one which maximizes
the growth rate (Hillert, 1972) or the inte-
gral rate of dissipation of Helmholtz
energy (Cahn, 1959). Solorzano and Purdy
(1984) put this latter hypothesis to the test,
and found a reasonable level of confirma-
S
V
F
min=
m
ch2s
ab
D
Figure 7-23.Schematic (plan)
view of a steady discontinuous
growth front, defining the
spacings, S, S
a, S
b, relevant to
the problem, and indicating the
vectorial balance of interfacial
energies,
s
a/b, s
a
0
/a, s
a
0
/b, at
the three-grain junction.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

561. tion for two well-characterized systems,
Al–22 at.% Zn and Mg–9 at.% Al.
Building upon the local approach of Hil-
lert and co-workers, Klinger et al. (1996)
have proposed a solution which leads to a
selection of the growth velocity without the
need for an optimization principle. This
treatment relies on the cooperative nature
of the growth of the aand blamellae, and
on the different characters of the interfaces
between the aand bphases and the parent
aphase: one is a grain boundary and the
other is an interphase boundary. Assuming
a definite composition for the bphase leads
to a Mullins (1957) equation for transport
along the a
0
/binterface determined by gra-
dients of curvature. Transport along the
a
0
/ais considered to be driven by concen-
tration gradients, according to Cahn’s
(1959) formulation. The condition at the
triple junction is taken as a local equilib-
rium, corrected for Gibbs–Thompson ef-
fects. The requirement that the aand b
growth velocities match in order to have a
cooperative growth process is sufficient to
select the spacing and velocity of the front.
This approach has the advantage of avoid-
ing the introduction of optimization princi-
ples, and it has been shown to accurately
describe some experimental results for the
Al–Zn system. It also predicts that steady-
state solutions with invariant front shapes
are possible within a limited range of driv-
ing forces, suggesting the possibility of
morphological instabilities outside this
range.
All of these theories of steady growth as-
sume that the motion of the interface is
continuous. This assumption is probably
acceptable when the growth front is well-
rounded, but there is substantial evidence
for faceted interfaces at low supersatura-
tions, as discussed in detail in the next sec-
tion. In the high-supersaturation regime,
the continuum models seem to apply with-
out modification; however, the theories
need to take into account the possibility of
lateral migration of the growth interface for
lower supersaturation. A recent treatment
by Klinger et al. (1997a) takes this into ac-
count by describing the motion of the inter-
face as intermittent.
7.5.3.3 Experimental Observations
The majority of experimental studies
have focused on microstructures developed
through steady isothermal growth. They
are therefore capable of description in
terms of the theoretical treatments of the
previous section. The types of information
accessible to experiment are:
a) quantities that may be derived from
conventional metallographic methods: av-
erage velocity vand spacing S;
b) chemical information obtained either
from averaging processes such as X-ray
diffraction measurements of lattice param-
eters of product lamellae or from high res-
olution microanalyses of the product;
c) structural information relating to the
growth interface, as obtained for example
by transmission electron microscopy.
In addition, for the test of theory we re-
quire solution thermodynamic data, inter-
facial energies and interfacial diffusion co-
efficients. Many alloy systems have been
investigated. In this section we will focus
on the systems Al–Zn and Mg–Al, for
which much of the necessary data are avail-
able (e.g., Rundman and Hilliard, 1967;
Cheetham and Sale, 1974; Hassner, 1974).
Yang et al. (1988) collected velocity and
spacing data for Al–Zn alloys, and for a
range of reaction temperatures and spac-
ings, as summarized in Figs. 7-24 and 7-25.
Figure 7-26 gives the average composition
of the a lamellae superimposed on the Al-
rich portion of the phase diagram. These
510 7 Transformations Involving Interfacial Diffusionwww.iran-mavad.com
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562. 7.5 Examples 511
composite figures include results from the
work of Ju and Fournelle (1985), Suresh
and Gupta (1986), Cheetham and Sale
(1974), Razik and Maksoud (1979), Wata-
nabe and Koda (1965), Gust et al. (1984)
and Yang et al. (1988). Perhaps the most
striking feature of these data is the extent
of residual supersaturation left in the a
phase behind the transformation front; a
significant amount of the total Helmholtz
energy for the transformation is stored in
the product phase.
This stored Helmholtz energy, first em-
phasized by Cahn (1959), must be esti-
mated in order to evaluate the driving force
for the transformation. This estimation is
readily performed using data of the type
shown in Fig. 7-26. This stored Helmholtz
energy provides a portion of the driving
force for subsequent discontinuous coars-
ening processes, which generally increase
the interlamellar spacing and deplete the
product ato (near) the solubility limit
(Yang et al., 1988).
High resolution microanalysis is capable
of providing further insights into the stored
Helmholtz energy term (Zie¸ba and Gust,
1998). The elemental trace of Fig. 7-27
was obtained from a STEM (scanning
transmission electron microscope) X-ray
microanalysis of a thin foil of Al–22 at.%
Zn transformed at 428 K. Measurements of
this type give both local and integral values
of the stored Helmholtz energy. In addition
to the information concerning the solute
concentration distribution in the aphase,
these profiles can be analyzed to yield
rather directly the product {sD
b
d} via Eq.
(7-23), (Porter and Edington, 1977; Solor-
Figure 7-24.Grain-boundary velocities for a range
of reaction temperatures and for several Al–Zn al-
loys (index DP denotes discontinuous precipitation).
The data are from studies by Yang et al. (1988), Ju
and Fournelle (1985) [1], Suresh and Gupta (1986)
[2], Cheetham and Sale (1974) [9], Razik and Mak-
soud (1979) [10], Gust et al. (1984) [18], and Wata-
nabe and Koda (1965) [19]. Reprinted with permis-
sion from Yang et al. (1988).
Figure 7-25.Interlamellar spacing measurements
for discontinuous precipitation (DP) products, from the same sources as for Fig. 7-24. Reprinted with per- mission from Yang et al. (1988).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

563. zano et al., 1984; Duly et al., 1994a, b).
However, in their studies of regular growth
in Mg–Al alloys, Duly et al. determined
that:
a) In certain cases, the concentration
profiles between parallel lamellae (similar
to Fig. 7-27) were well described by
Cahn’s equation with constant values of
C*
aband ÷
–
a.
b) However, within a given nodule, the
value of (a/S
a
2) was generally not constant,
varying by more than an order of magni-
tude. This quantity is proportional to the
instantaneous interfacial velocity, and its
variation is taken to mean that, on an
atomic length scale, the interface velocity
is irregular. On a larger length scale, typi-
cally 0.1 µm, an average velocity can be
defined that is consistent with the steady
diffusion analysis of Cahn.
A major implication of these findings is
that the transformation interface moves by
a lateral growth mechanism, resulting in an
average velocity much less than the instan-
taneous velocity accompanying the pas-
sage of a growth step. It is here that the
recent studies of Shiflet and his co-workers
(Fonda and Shiflet, 1990; Fonda et al.,
1998) on Cu–Ti alloys are relevant. Their
results indicate a strong and consistent ten-
dency for the transformation front to facet,
particularly at low undercoolings, and this
implies a lateral displacement process,
consistent with the findings of Duly et al.
512 7 Transformations Involving Interfacial Diffusion
Figure 7-26.Average zinc concentrations in the a
phase for discontinuous precipitation (DP) and for
discontinuous coarsening (DC) reactions in various
Al–Zn alloys. Reprinted with permission from Yang
et al. (1988).
Figure 7-27.STEM microanalysis
of an a lamella in Al–22 at.% Zn,
formed at 428 K. After Solorzano and Purdy (1984).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

564. 7.5 Examples 513
(1994a, b). At lower reaction temperatures,
the growth interfaces in Cu–Ti become
smoothly curved. This is attributed to the
rapid accumulation of growth ledges with
increasing undercooling.
The smoothly curved interfaces and
steady migration rates seen in high resolu-
tion in situstudies of discontinuous precip-
itation in Al–Zn (see Fig. 7-28, Tashiro
and Purdy, 1989) suggest that, at lower
temperatures, the interface can be consid-
ered to move continuously. Indeed, the
interface shapes observed could be ration-
alized in terms of a local balance between
chemical (evaluated from microanalyses),
capillary and friction forces, realized at
every point on the transformation front. It
therefore appears that at some stage an ef-
fective transition occurs where the accu-
mulation of growth ledges is sufficient to
mimic the normal migration of the front,
and the classical theories of Cahn and Hil-
lert then hold to a good approximation.
There is a further set of experimental
data on discontinuous precipitation that re-
lates to the effect of an applied stress on the
transforming system. Sulonen (1964a, b)
showed that in some alloy systems, fronts
with normals parallel to the tensile axis
moved more slowly, and those with nor-
mals perpendicular to the stress axis more
quickly than those in unstressed alloys. In
other alloy systems, the opposite behavior
was found. The effect for Cu–Cd alloys is
shown in Fig. 7-29. Sulonen proposed an
Figure 7-28.Observed and calculated interface
shapes for an alamella formed at 400 K in Al–
22 at.% Zn. The numbers on the curves represent
different amounts of Helmholtz energy lost owing to
continuous precipitation in the parent phase.
Figure 7-29.Effect of an applied tensile stress on
the rates of growth of the discontinuous product in Cu–Cd alloys (1 kp/mm
2
≈9.81¥10
6
N/m
2
). Re-
printed with permission from Sulonen (1964b), copyright Pergamon Press.www.iran-mavad.com
+ s e l ∫'4 , kp e r i ∫&s ! 9 j+ N 0 e

565. explanation based on the elastic interaction
between the applied and coherency (solute)
stress fields. Hillert (1972) indicated how a
quantitative treatment of such an effect
could be formulated. Sulonen’s result and
Hillert’s analysis have been widely quoted
in support of the existence of a solute (co-
herency) field in front of the discontinuous
transformation front, even at temperatures
where the calculated diffusion distance is
negligible.
Dryden and Purdy (1990) reconsidered
the problem, and included the possible ef-
fect of a volume misfit in the transformed
region. It was found, in agreement with
Hillert, that the sign of the effect could be
predicted on the basis of an elastic solute
field interaction; however, the predicted ef-
fects on interface velocities are too small
by four orders of magnitude. The more
likely cause of the coupling is found in the
plastic response of the dead-loaded speci-
men. The reduction in Helmholtz energy of
the loading device results in a virtual force
on the transformation interface, as dis-
cussed in Sec. 7.4.3.2. For the case of inter-
faces with normals parallel to the x, y, and
z(tensile) axes, these virtual forces are re-
lated by
(7-28)
where p
x, p
y, and p
zare forces per unit
area, Zis the applied tensile stress,
eis the
stress-free strain in the transformed vol-
ume, and
Wis an alignment parameter
which accounts for the possibility that the
lamellae in transformed regions bounded
by xand yinterfaces are aligned with re-
spect to the z axis.
It is found that the virtual forces so de-
rived, in conjunction with independently
derived grain-boundary mobilities, are ca-
pable of explaining the sign and magnitude
of Sulonen’s data for Cu–Cd alloys (Fig.
pp p
Z
xy z== =−
+⎛
⎝
⎞
⎠
+
13
22
13
W
W
e
()
7-29), and give the correct sign of the ef-
fect for the other five systems investigated,
for which no quantitative data were ob-
tained. We conclude that it is the plastic (or
creep) response of the specimen which is
more plausibly coupled to the migration
rates of differently oriented transformation
interfaces.
7.5.4 Interface Migration in Multilayers
Multilayers have been utilized for criti-
cal experiments since the seminal work of
Hilliard and his co-workers (1954). They
provide an ideal tool for the investiga-
tion of the thermodynamics and kinetics of
heterogeneous systems far from equilib-
rium. Much of the work to date has been
concerned with the approach to equilib-
rium via bulk diffusion. The recently de-
veloped techniques for precisely controlled
growth (e.g., molecular beam epitaxy)
prompt the exploration of the approach to
equilibrium via interfacial diffusion and
migration. The theoretical investigations
summarized in this section deal with pos-
sible effects, many of which are still to be
observed.
Several classes of problems can be de-
veloped, beginning with an A–B multi-
layer grown on a bicrystalline substrate
such that a grain boundary penetrates the
whole structure and provides a possible
fast diffusion path. The simplest case oc-
curs when A and B are fully miscible.
Homogenization can then occur by grain
boundary motion, either via a cooperative
mechanism, or by a “fingering solution” at
the former A/B interfaces as illustrated in
Figure 7-30 (Klinger et al., 1997a). In each
case, the shape of the moving boundary, as
well as its velocity and the concentration
profile left in its wake, can be computed in
terms of driving forces, interface energies
and diffusion coefficients.
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566. 7.5 Examples 515
The next situation to be considered is
when A and B are reactive, and form a stoi-
chiometric compound
w(or, conversely,
when such a compound decomposes to
form A and B). Again, for this case, the ve-
locity of the cooperative moving front for
reaction (or dissolution) can be computed
as a function of the energies and mobilities
involved (Klinger et al., 1997c). A solution
involving a reaction product layer at the
interface rather than a cooperative growth
front can also be considered (Fig. 7-31),
and this may help to explain certain pecu-
liarities found in kinetics in reactive multi-
layers, in terms of diffusion barrier effects
(Klinger et al., 1998; Emeric, 1998).
These theoretical investigations, many
still awaiting experimental confirmation,
suggest a wide variety of possible kinetic
paths toward thermodynamic equilibrium
(involving pattern and velocity selection
as well as morphological instabilities) in
well-controlled systems. In these systems,
a number of classical hypotheses such as
that of local equilibrium could be checked
quantitatively. It is suggested that the
search for such interface-mediated structu-
ral evolution in controlled multilayered
structures can play a key role in the deeper
understanding of the more frequently en-
countered phenomena described in the rest
of this chapter.
Figure 7-30.Illustrating three different possibilities for the discontinuous homogenization of a monophase
multilayer containing a mobile grain boundary, assumed initially to bisect the multilayer. Situations correspond-
ing to (a) the steady state motion of the boundary; (b) an initially sinusoidal instability of the grain boundary,
and (c) a “fingering” instability of the moving boundary, after Klinger et al. (1997b).
Figure 7-31.Schematic representation of the pos-
sible reactions of A and B to form a stoichiometric product phase w. (a) A cooperative reaction at a sin-
gle front; (b) the growth of a product layer at the interface (after Klinger et al., 1997c).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

567. 7.6 Conclusions
In this chapter we have been concerned
with those phase transformations that de-
pend on diffusion within a transformation
front. The massive transformation, chemi-
cally-induced grain boundary migration,
discontinuous precipitation and transfor-
mations in multilayers represent different
and distinct facets of the general class of
interface-diffusion controlled transforma-
tions.
In each case, the description of the trans-
formation in terms of a balance of forces on
the moving interface has proved a unifying
concept. For the massive transformation, it
is believed that the major forces at play are
the chemical driving force and the oppos-
ing frictional forces. In the description of
chemically-induced grain boundary migra-
tion, especially at higher temperatures, an
elastic (coherency) term due to a gradient
of misfitting solute atoms in the parent
grain must be added to the other two types
of force. For discontinuous precipitation,
capillary forces must be introduced to op-
pose the chemical driving force. We have
also considered transformations in multi-
layer heterostructures, where capillary
terms must be accounted, and where inter-
facial diffusion is often driven by interfa-
cial curvature.
The description offered has been pre-
dominantly thermodynamic, rather than
deeply mechanistic. Much remains to be
learned about the mechanisms of accom-
modation of diffusing atoms within the
transformation interface, about the cou-
pling of the driving forces with interface
response, and about the interrelationships
between interfacial structure and kinetics.
It has become increasingly clear that many
transformation interfaces are faceted, and
remain faceted during migration. In such
cases, the classical models for diffusional
transformations will need to be reformu-
lated, and this is only beginning to take
place.
The thermodynamic structure remains a
valuable synthetic framework within which
to place advances in the understanding of
this fascinating field.
7.7 References
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Chen, J. K., Purdy, G. R., Weatherly, G. C., Kroupa,
A. (1998), Metall. Mater. Trans. A 29, 2049.
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Cottrell, A. H. (1953), Dislocations and Plastic Flow
in Crystals, Oxford: Oxford University Press.
Dahmen, U. (1981), Scripta Metall. 15, 77.
Delaey, L. (2001), in: Phase Transformations in Ma-
terials:Kostorz, G. (Ed.). Weinheim: Wiley-VCH,
Chap. 9.
Delage, S. (1998), Ph.D. thesis, University of Paris V.
den Broeder, F. J. A. (1972), Acta Metall. 20, 319.
Dryden, J. R., Purdy, G. R. (1990), Acta Metall. 38,
1255.
Duly, D., Bréchet, Y. (1994), Acta Metall. Mater. 42,
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518 7 Transformations Involving Interfacial Diffusionwww.iran-mavad.com
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570. 8 Atomic Ordering
Gerhard Inden
Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Federal Republic of Germany
List of Symbols and Abbreviations  . . . . . . . . . . . . . . . . . . . . . . . . . . 521
8.1 Introduction................................ 523
8.2 Definition of Atomic Configurations................... 523
8.2.1 Configurational Variables  . . . . . . . . . . . . . . . . . . . . . . . . . . 523
8.2.2 Point Variables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
8.2.3 Point Correlation Functions and Point Probabilities  . . . . . . . . . . . . 526
8.2.4 Pair Variables, Correlation Functions, and Probabilities  . . . . . . . . . . 527
8.2.5 Generalized Cluster Variables, Correlation Functions, and Probabilities . . 529
8.2.6 Short-Range-Order (sro) Configurations – 
Long-Range-Order (lro) Configurations  . . . . . . . . . . . . . . . . . . 529
8.3 The Existence Domain and Configuration Polyhedron......... 531
8.3.1 F.C.C. Structure, First Neighbor Interactions  . . . . . . . . . . . . . . . . 532
8.3.2 F.C.C. Structure, First and Second Neighbor Interactions  . . . . . . . . . 535
8.3.3 B.C.C. Structure, First and Second Neighbor Interactions  . . . . . . . . . 537
8.4 Ground States............................... 538
8.4.1 Pair Interactions  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
8.4.1.1 Ground State Energies  . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
8.4.1.2 F.C.C. Structure, First Neighbor Interactions  . . . . . . . . . . . . . . . . 540
8.4.1.3 F.C.C. Structure, First and Second Neighbor Interactions  . . . . . . . . . 543
8.4.1.4 B.C.C. Structure, First and Second Neighbor Interactions  . . . . . . . . . 544
8.4.1.5 Energy Minimum at Constant Composition . . . . . . . . . . . . . . . . . 544
8.4.1.6 Canonical Energy of lro States  . . . . . . . . . . . . . . . . . . . . . . . 546
8.4.1.7 Relevant Literature  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
8.4.2 Effective Cluster Interactions (ECIs)  . . . . . . . . . . . . . . . . . . . . 548
8.5 Phase Equilibria at Finite Temperatures................. 551
8.5.1 Cluster Variation Method  . . . . . . . . . . . . . . . . . . . . . . . . . . 551
8.5.2 Calculation of Phase Diagrams with the CVM  . . . . . . . . . . . . . . . 554
8.5.3 Phase Diagram Calculation with the Monte Carlo Method  . . . . . . . . . 554
8.5.4 Examples of Prototype Diagrams  . . . . . . . . . . . . . . . . . . . . . . 555
8.5.4.1 F.C.C. Structure, First Neighbor Interactions  . . . . . . . . . . . . . . . . 555
8.5.4.2 F.C.C. Structure, First and Second Neighbor Interactions  . . . . . . . . . 557
8.5.4.3 B.C.C. Structure, First and Second Neighbor Interactions  . . . . . . . . . 558
8.5.4.4 Hexagonal Lattice, Anisotropic Nearest-Neighbor Interactions  . . . . . . 558
8.5.5 The Cluster Site Approximation (CSA) . . . . . . . . . . . . . . . . . . . 560
8.6 Application to Real Systems........................ 561
8.6.1 The Au–Ni System  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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571. 8.6.2 The Thermodynamic Factor of Ordered Phases . . . . . . . . . . . . . . . 565
8.6.2.1 The B.C.C. Fe–Al System  . . . . . . . . . . . . . . . . . . . . . . . . . 565
8.6.2.2 The F.C.C. Ni–Al System . . . . . . . . . . . . . . . . . . . . . . . . . . 567
8.6.3 Ternary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
8.6.3.1 B.C.C. Fe–Ti–Al  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
8.6.3.2 B.C.C. Ferromagnetic Fe–Co–Al  . . . . . . . . . . . . . . . . . . . . . 570
8.6.4 H.C.P. Cd–Mg  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
8.6.5 Concluding Remarks  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
8.7 Appendix.................................. 575
8.8 Acknowledgements............................. 578
8.9 References................................. 578
520 8 Atomic Orderingwww.iran-mavad.com 
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572. List of Symbols and Abbreviations 521
List of Symbols and Abbreviations
a, a
1, … basic vectors of the structure
a
a CVM exponents
b, b
1, … basic vectors of the structure
c, c
1, … basic vectors of the structure
d distance of hyperplane to the origin
E
˜
part of internal energy that depends only on the configurational variables
F Helmholtz energy
i number identifying the type of atom
k number identifying a neighbor distance
K number of constituents
k
B Boltzmann constant
m
a number of a-clusters per point
N number of lattice sites
n, m number identifying a lattice site
n normal vector
N
i number of atoms i
N(
s
r) number of r-clusters with occupation  s
r
N
r number of equivalent r-site clusters
p
n
(i) site-occupation operator for atom ion site n
p
ij
nm
occupation operator for atom pair i, jon sites n, m
r number of lattice points within a cluster
S entropy
T temperature
U, U
˜
internal energy, grand canonical internal energy
V volume
V
ij
nm pair energy of atoms i, jon sites n, m
W number of possible arrangements of clusters formed for given values of the
correlation functions
W
ij
(k) pair exchange energy of k-th neighbor atoms Iand J
W
(k)
binary case: indices ijomitted
W
ij…l
12…r r-site cluster exchange energy correction term
V
n cluster expansion coefficient
w
n
iÆj transition probabilities
x hyperplane position vector
x
v vertex vector
x
i mole fraction of atom of type i
z
(1…r)
coordination number of r-site clusters
a number identifying a cluster
a
ij
n–m
Cowley–Warren sro parameters
m
i chemical potential of atoms i
n dimension of configurational spacewww.iran-mavad.com
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573. n
tot, n
p total number of configurational probabilities, number of configurational prob-
abilities related by partial summations
z
1, z
2, z
3lro parameters
r(s
r) N( s
r)/N
r
r
ij
nm
probability of having atoms i, jon sites n, m
s vector with site operators as components
s
n site operator or spin variable
t
k values of the site operator
j arbitrary function
F
i,F energy variables, defining dual space of the configurational variables, their
vector hyperplane
W grand potential
CVM cluster variation method
CSA cluster site approximation
EC cluster expansion
ECI effective cluster interaction
lro long-range order
n–m neighbor distance between sites n, m(equivalent to number k)
MC Monte Carlo simulation
sro short-range order
1k
B-unit = 1k
BK = 13.8 ¥10
–24
J = 8.6 ¥10
–2
meV
522 8 Atomic Orderingwww.iran-mavad.com
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574. 8.2 Definition of Atomic Configurations 523
8.1 Introduction
Ordered structures occur frequently in
solid solutions and exhibit interesting
physical and mechanical properties. This
explains the continuous interest in ordering
reactions by theoreticians and experimen-
talists over recent decades. There are some
excellent overview articles and books (e.g.,
Ducastelle, 1991; Turchi and Gonis, 1994;
de Fontaine, 1979, 1994; Khachaturyan,
1978) on the theoretical aspects of this sub-
ject, while the experimental and materials
science aspects can be found in articles by
Kear et al. (1970), Warlimont (1974), Koch
et al. (1985), Stoloff et al. (1987), Liu et al.
(1989, 1992), and Whang et al. (1990).
The ordered phases can be classified
under the family of intermetallic phases.
As such they have attracted interest in the
development of materials for special appli-
cations (e.g., materials for use at high tem-
peratures). Thus there is an increasing de-
mand for quantitative descriptions of the
variations in their properties. Furthermore,
materials with practical relevance rarely
consist of binary systems. Treatments of
multicomponent systems are thus espe-
cially needed. The most fundamental prob-
lems that need to be solved at the start of
any optimization of materials involve the
type of ordered structure, the variation of
the atomic distribution with temperature
and composition, and the phase equilibria.
This chapter is an updated version of the
previous contribution by Inden and Pitsch
(1991). The tutorial aspects have been
maintained as much as possible. It is for
this reason that the basic ideas are still 
presented in terms of a pair interaction
scheme, which can only provide prototype
results. An up-to-date treatment of real
systems requires more sophisticated ap-
proaches that take into account previously
neglected but very important physical ef-
fects. First-principles calculations of total
energies, of lattice relaxations and of local
relaxations were still in their infancy ten
years ago, but such theoretical calculations
are now available. They will be discussed
in relation to the treatment of real alloys.
The variety of materials exhibiting or-
dering is too great to be dealt with com-
pletely. The present chapter will therefore
be limited to metallic substitutional alloys,
despite the fact that the techniques dis-
cussed have also been used extensively in
the field of interstitial alloys, carbides, ni-
trides, oxides, and semiconductor systems.
The chapter is organized as follows. In
Sec. 8.2, a general formalism for describ-
ing and characterizing atomic configura-
tions in multicomponent substitutional al-
loys is presented. The concept of correla-
tion functions as independent variables for
the definition of the configurations is intro-
duced and used in Sec. 8.3 to derive exis-
tence domains; they limit the range of nu-
merical values that can be scanned by the
correlation functions for topologically ex-
isting configurations. These existence do-
mains are used in Sec. 8.4 to determine the
ground states. In Sec. 8.5, equilibrium at fi-
nite temperatures is discussed in terms of
the cluster variation method (CVM) and
the Monte Carlo simulation (MC). Finally,
in Sec. 8.6, the application to real systems
is discussed.
8.2 Definition of Atomic
Configurations
8.2.1 Configurational Variables
Let us consider a crystalline system with
Nlattice sites and Kconstituents i=A,
B, … . Defining the number of atoms of
type iby N
igives Â
i
N
i=Nand the mole
fractions x
i=N
i/N. The distribution of thewww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

575. atoms on the lattice sites defines the con-
figuration.
In order to specify a configuration, we
need an operator which identifies unequiv-
ocally the atomic species on an arbitrary
site n. A convenient means of identifica-
tion is to associate an integral number with
each constituent and to define a site opera-
tor 
s
n, which takes these integral values
corresponding to the constituent on site n.
Different choices are possible, e.g.,
Any configuration is then specified by the
vector 
s=(s
1,s
2,…,s
N). In total, there
are K
N
different configurations. For binary 
alloys, the choice  was first sug-
gested by Flinn (1956). The operator 
s
nis
sometimes called the spin variablebecause
of its correspondence with the Ising model
for binary alloys if we take 
s
n= ±1. Any
function of 
s
nincluding s
nitself is called a
configurational variable.
Because Nis a very large number, it is
not possible to handle such a large amount
of information. Therefore, we are forced to
work with a reduced amount of information
by considering the configurations of much
smaller units called clusters. A cluster is 
defined by a set of lattice points 1, 2, …,r,
and a configuration on this cluster is given
s
n
x
x
=
B
A
−
⎧
⎨
⎩
or generally
for = A
for = B
for =
t
t
t
1
2
≠
K
i
i
iK
⎧
⎨
⎪
⎩
⎪
s
n
KK
KK
KK
=oror
0
1
2
1
1
2
3
2
1
2
1
0
1
2
1
2
≠≠
≠
≠
−
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⎧
⎨
⎪
⎪
⎩
⎪
⎪
−⎛
⎝
⎞
⎠
−
−−+⎛
⎝
⎞ ⎠
⎧
⎨
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
()
by 
s
r=(s
1,s
2,…,s
r). The smallest clus-
ter is a point, the next one a pair, then a
triplet, and so on. On an r-site cluster there
are K
r
configurations, a much smaller num-
ber than K
N
. The configurations of the N-
point system can then be classified into
groups with the same number of clusters
N(
s
r) with the configuration s
r. Instead of
N(
s
r), it is preferable to work with the re-
duced number 
r(s
r)=N(s
r)/N
r, where N
r
is the number of equivalent r-site clusters
contained in the system. These fractions
are called cluster probabilities. They spec-
ify the configuration in the r-point cluster
approximation and constitute the most im-
portant configurational variables. The ap-
proximation depends on the size of the
largest cluster taken into account. For a
three-dimensional (3-dim.) lattice, it is
necessary to include at least one 3-dim.
cluster, otherwise the topological connec-
tion of the clusters for space filling cannot
be taken into account correctly. For in-
stance, if only pairs are considered, it is not
possible to distinguish a 3-dim. configura-
tion from a 2-dim. Bethe lattice with the
same coordination number.
So far rdefines a particular set of points
1, 2, …,r, and N
ris the number of clusters
having the same orientation in space, thus
differing from this particular set by a trans-
lation in the lattice, or N
r=N. In this in-
stance, we speak of oriented clustersand
oriented-cluster probabilities. The N-point
system usually exhibits more symmetry
elements than translation, and clusters of
different orientation then become equiva-
lent. Bringing these clusters together gives
N
r>N.
Suppose that an r-point cluster has been
selected for the description of the config-
urations. Then the thermodynamic func-
tions derived, for example, with CVM, de-
pend on the probabilities of this cluster and
also on the probabilities of all subclusters.
524 8 Atomic Orderingwww.iran-mavad.com 
+ s e  l ⎨'4 , kp e  r  i ⎨&s ! 9 j+  N   0 e

576. 8.2 Definition of Atomic Configurations 525
Therefore, the total number of configura-
tional probabilities 
n
totadds up to
because for a point, there are Kchoices of 
an element and  choices of a point in 
the cluster, for a pair, there are K
2
choices 
of two elements and  choices of a pair 
of points in this cluster, and so on. How-
ever, these 
n
totprobabilities are not all in-
dependent: if the probabilities of the largest
cluster are given, then the probabilities of
all subclusters can be derived by partial
summation, e.g., for a 3-point cluster
(8-1)
The number of these partial summations in
Moreover, the sum of the probabilities of
the largest cluster is equal to 1 by defini-
tion. Therefore the number of independent
probabilities is
n= n
tot– n
p–1= K
r
–1
that is, 
nis the dimension of the configura-
tional space. When rand Kincrease, the
difference 
n
tot–n
pbecomes very large.
Of course any choice of 
nprobabilities
out of the whole set 
n
totmay serve as a set
of independent configurational variables.
A convenient choice is any selection of 
n
probabilities out of the K
r
probabilities of
the largest cluster. This is usually done in
Kikuchi’s natural iteration method (Kiku-
chi and Sato, 1974). It is necessary, how-
n
p=
r
K
r
K
r
r
K
r
12 1
21⎛
⎝
⎞
⎠
+
⎛
⎝
⎞
⎠
+…+
−
⎛
⎝
⎞
⎠ −
rsss
rs s s s s
ss
(, , )
(, , , , , )
123
1234
4
=
…
∑ …
r
r
r
2
⎛ ⎝
⎞ ⎠
r
1
⎛ ⎝
⎞ ⎠
n
tot=
=
r
K
r
K
r
r
K
K
r
r
12
11
2⎛
⎝
⎞
⎠
+
⎛
⎝
⎞
⎠
+…+
⎛
⎝
⎞
⎠
+−()
ever, to introduce Lagrange parameters in
the minimization of Gibbs energy in order
to take account of the 
n
pconsistency rela-
tions of Eq. (8-1), and it is not straightfor-
ward to make use of efficient minimization
algorithms. Therefore, effort has been con-
centrated in defining a set of independent
variables (or a basis) in the 
n-dim. space
such that all the configurational variables
can be expressed in terms of this basis. For
binary alloys, Sanchez and de Fontaine
(1978) used multisite correlation func-
tions as an extension of the pair correlation
functions introduced by Clapp and Moss
(1966). An extension to multicomponent
systems was first proposed by Taggart
(1973) using the spin concept, after which
Sanchez et al. (1984) suggested Chebychev
polynomials as a basis of the configura-
tional space. In the following another basis
is developed by a method that is equivalent
to the one by Taggart. This basis is simpler
and better suited to numerical applications
than that by Sanchez et al. (1984).
The procedure is as follows. First, we de-
fine a basis in the space of point variables.
This yields (K– 1) functions of 
s
n. Then,
we consider the (K
2
– 1)-dim. space of the
two-point variables and define a basis by
taking products of the (K– 1) basic point
variables, and so on. This explains why the
most important step is an appropriate
choice of the basis for point variables.
8.2.2 Point Variables
In the case of point variables, rcorre-
sponds to one point, and the vector 
sre-
duces to one element 
s
n. In order to define
a basis of configurational point variables,
in particular for the point probabilities
r(s
n), it is helpful to introduce a second
operator p
n
(i)which allows us to count the
number of sites nwith the same type of
atom ifor taking averages. This operatorwww.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

577. p
n
(i)is called the site-occupation operator
(Clapp and Moss, 1966) and is defined as
follows:
Because igoes from 1 to Kwe have Kop-
erators. Using this defintion we can imme-
diately write the following equations:
(8-2)
where the upper indices signify powers 
of 
s
nor t
i. This notation will be used
throughout this chapter. To distinguish an
upper index from a power, the index will be
put in parentheses, except if it is a double
or multiple index, which cannot be con-
fused with a power. Let us call Mthe ma-
trix of this system of equations. Its deter-
minant is the van der Monde determinant,
which is given by
Because all the 
t
iare different numbers,
this determinant is different from zero, and
the equations are linearly independent.
This confirms that the K– 1 functions {
s
n,
s
n
2,…,s
n
K–1} are linearly independent.
Det =M
ji
K
i
r
ji
=+ =
∏∏ −
11
()tt
=
111 1
123
1
2
2
2
3
22
1
1
2
1
3
11
1
2
3
…
…
…
…
…
ttt t
ttt t
ttt t
K
K
KKK
K
K
n
n
n
n
K
p
p
p
p
⋅⋅⋅ ⋅
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⋅
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
−−− −
()
()
()
()
1 1
1
1
2
1
2
1
1
1
2
1
=
=
=
=
or
i
K
n
i
n
i
K
in
i
n
i
K
in
i
n
K
i
K
i
K
n
i
n
n
n
K
p
p
p
p
=
=
=
−
=
− −
∑
∑
∑
∑
⎧
⎨
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
()
()
()
()
st
st
st
s
s
s
≠ ≠
⎜⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
p
in
n
i()=
if an atom of type   occupies site 
otherwise
1
0
⎧
⎨
⎩
DetM≠0 and so we can invert the ma-
trix Mand define R=(R
ik)=M
–1
. From
Eq. (8-2) we arrive at
(8-3)
This is the relation that connects the site-
occupationoperators with the site opera-
tors. Because all the elements of the first
row of Mare unity, the inverse matrix R
exhibits the following properties: the ele-
ments of the first column of Radd up to
unity and the elements of all other columns 
add up to zero, i.e.,
8.2.3 Point Correlation Functions
and Point Probabilities
So far we have considered the point var-
iables for one arbitrary point nout of the N
lattice points of the crystal. As mentioned
before, we want to reduce the number of
parameters describing an atomic configura-
tion by considering averages over equiva-
lent clusters, or in this case, points. The
space group of the structure will define the
equivalence of points. We may enumerate
the classes of equivalent points by 1, 2, …
and define the number of lattice points in
each class by N
(1)
, N
(2)
,…,N
(L)
. The aver-
age of an arbitrary function 
j(s
n,s
n
2,…)
i
K
ik k
R
=
∑
1
1
=d.
pR
pR
pR
p
p
p
p
R
n
k
K
kn
k
n
k
K
kn
k
n
K
k
K
Kkn
k
n
n
n
n
K
n
n
n
K
()
()
()
()
()
()
()
1
1
1
1
2
1
2
1
1
1
1
2
3 2
1
1
=
=
=
or =
=
−
=
−
=
−
−
∑
∑
∑
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
s
s
s
s
s
s
≠
≠ ≠
⎟⎟
⎟
⎟
⎟
526 8 Atomic Orderingwww.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

578. 8.2 Definition of Atomic Configurations 527
of the point variables s
n,s
n
2, … is defined
by
The index n now stands for one representa-
tive point out of the corresponding class.
The averages of the site-occupation opera-
tors p
n
(i)yield the probabilities of finding an
atom A, B, …, Kon a point n:
In the limit of NÆ∞, this average cor-
responds to the thermodynamic site-occu-
pation probability. The (K– 1) functions
·
s
n
K–1Òare called point correlation func-
tions. These functions constitute the basis
of the configurational space of point vari-
ables, and Eq. (8-4) yields the components
of the point probabilities with respect to
this basis.
8.2.4 Pair Variables, Correlation Func-
tions, and Probabilities
Consider a two-point cluster r=n,m. We
can immediately introduce a pair-occupa-
tion operator that takes the value 1 if an
atom of type ioccupies site n , and if type j
occupies site m. This operator is simply the
product of the two previously introduced
site-occupation operators p
ij
nm
=p
n
(i)p
m
(j).
Using the expressions of the site operators,
Eq. (8-3), we obtain
The pair probabilities 
r
ij
nm
are obtained by
taking the average of the operator p
ij
nm
over
all equivalent pairs in the crystal. It is
worth mentioning that the pairs may be
non-equivalent if they differ by their orien-
pRR
nm
ij
k
K
h
K
ih jkn
h
m
k=
==
−−
∑∑
11
11
ss
rst s()
,,
()
ni n
i
k
K
ikn
k
pR
iK
== =
=(8-4)
〈〉 〈 〉
…
=
−
∑
1
1
1
〈…〉 …
=
∑js s js s(, ,) (, ,)
()
()
nn n
s
N
ss
N
n
2
1
2 1
=
tation, i.e., if nand mare non-equivalent
points. In this instance, we speak of oriented pair probabilities. In the opposite
case, we speak of isotropicpair probabil-
ities:
(8-5)
The pair probabilities can thus be ex- pressed in terms of the already introduced point correlation functions, written here in the form ·
s
n
h–1s
m
k–1Òwith h=1 and k= 2, 3,
…, K(or k=1 and h = 2, 3, …, K) together
with a second set of functions, ·
s
n
h–1s
m
k–1Ò,
with h, k= 2, …, Kcalled pair correlation
functions. The number of pair correlation
functions is (K–1)
2
if nand mare non-
equivalent sites. The total number of point and pair correlation functions adds up to 2(K–1)+(K–1)
2
=K
2
– 1. This is exactly
the dimension of the configurational space if we assume the pair to be the basic clus- ter. Point and pair correlation functions to- gether constitute a basis in this instance.
By virtue of the relation
Â
j
p
n
(i)p
m
(j)=p
n
(i)
the pair probabilities are consistent with the point probabilities.
For the purposes of illustration, let us
consider two examples:
Binary alloy
For a binary alloy, K= 2 and 
t
1= 1 for
the first element A, and 
t
2= –1 for the sec-
ond element B. The point probabilities fol-
low from
which yields
111
11
1
2
s
n
n
n
p
p
⎛
⎝
⎞
⎠ −
⎛
⎝
⎞
⎠
⎛
⎝
⎜
⎞
⎠
⎟=
()
()
MR=   and =
11
11
1
2
11
11−
⎛
⎝
⎞
⎠
−
−−
−
⎛
⎝
⎞
⎠
r
ss
nm
ij
n
i
m
j
k
K
h
K
ih jk
n
h
m
kpp RR==〈〉
×〈 〉
==
−−
∑∑
() ( )
11
11www.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

579. and
and consequently
(8-6)
For the pair probabilities, we obtain, ac-
cording to Eq. (8-5),
or
r
r
r
r
r
r
r
r
s
s
ss
nm
nm
nm
nm
nm
nm
nm
nm
n
m
nm
AA
AB
BA
BB
11
12
21
22
=( 8-7)
=
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
−−
−−
−−
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
〈〉
〈〉
〈〉
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
1
4
1111
1111
1111
1111
1
rs s
nm
ij
kh
ih jk n
h
m
k RR=
==
−−
∑∑ 〈〉
1
2
1
2
11
rs
rs
srr
nn
nn
nnn
()
()
() ( )()
()
11
2
21
2
12
1
1
=
=
and =
+〈 〉
−〈 〉
⎧
⎨
⎩
〈〉 −
p
p
p
p
n
n
n
n n
A
B
==
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
()
()
1
2
1
2
11
11
1
s
Ternary alloy
For a ternary alloy, K= 3 and 
t
1= 1 for
element A, 
t
2= 0 for B, and t
3= –1 for C.
In this instance, the point variables follow
from 
and
According to Eqs. (8-2) and (8-5), we ob-
tain
(8-8)
For the pair probabilities, we obtain, ac-
cording to Eq. (8-5),
〈〉 −
〈〉 +
〈〉+〈〉
−〈 〉
−〈 〉+〈 〉srr
srr
rr s s
rr s
rr s s
nnn
nnn
nn n n
nn n
nn n n=
=
==
==
==
A
B
C
() ( )
() ( )
()
()
()
()
()
()
13
213
11
2
2
21
2
2
31
2
2
22
R=
1
2
011
20 2
011
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
M=
11 1
10 1
10 1
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
528 8 Atomic Ordering
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
nm
nm
nm
nm
nm
nm
nm
nm
nm
nm
nm
nm
nm
nm
nm
nm
nm
nm
AA
AB
AC
BA
BB
BC
CA
CB
CC
11
12
13
21
22
23
31
32
33
==
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
1
4
000011 10 11
022000022
000011011
000202202
404000404
000202202
000011011
022000022
000011011
−−
−−
−−
−−
−−
−−
−−
−−
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟⎟
⎟
〈〉
〈〉
〈〉
〈〉
〈〉
〈〉
〈〉
〈〉
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
1
2
2
2
2
22
s
s
s
ss
ss
s
ss
ss
n
n
m
nm
nm
m
nm
nm
(8-9)www.iran-mavad.com 
+ s e  l ⎨'4 , kp e  r  i ⎨&s ! 9 j+  N   0 e

580. 8.2 Definition of Atomic Configurations 529
8.2.5 Generalized Cluster Variables,
Correlation Functions, and Probabilities
The point and pair variables are the sim-
plest examples of the general cluster vari-
ables. Consider an oriented cluster defined
by rpoints. As before, we introduce a clus-
ter-occupation operator by the product of
site-occupation operators p
1
(i
1)
p
2
(i
2)
…p
r
(i
r)
.
The cluster correlation functions are the
averages over all equivalent clusters in the
crystal. Thus the result for the cluster prob-
abilities is
(8-10)
The parameters  include the r -
point correlation functions (point, pair,
etc.) which provide the characterization of
the configuration by means of r-site clus-
ters, for example, the point correlation
functions are coded in the form ·
s
mÒ=
·
s
1
0…s
1
m
…s
r
0Ò, ·s
2
m
Ò=·s
1
0…s
2
m
…s
r
0Ò,
etc. and, correspondingly, for the pair cor-
relation functions, they are coded ·
s
ms
nÒ=
·
s
1
0…s
1
m
s
1
n
…s
r
0Ò, etc. The r-site cluster
correlation functions are obtained when
k
n≠1 for all values of k
n. If all the rsites
are not equivalent, the total number of 
r-point correlation functions is (K–1)
r
.
The total number of correlation functions
N
crequired for an r-point cluster treatment,
including all the subcluster correlation
functions, is
This is the dimension of the configura-
tional space, and the whole set of cluster
N
r
K
r
K
r
r
KK K
rrr
c
=
==
1
1
2
1
11111
2⎛
⎝
⎞
⎠
−+
⎛
⎝
⎞
⎠
−+…
+
⎛
⎝
⎞
⎠
−−+−−
() ()
()( )
n
r
n
k
n
=
−
∏
1
1
s
r
ss s
12 1 2
11 1
1
1
2
1 1
12 12
12
11 2 2
12
…
…
== =
−− −
〈…〉
……
×〈 … 〉
∑∑ ∑
r
ii i ii
r
i
k
K
k
K
k
K
ik i k ik
kk
r
k
r r
r
rr
r
pp p
RR R
=
=
() ( ) ()
correlation functions constitutes a basis of
this configurational space. If all sites are
equivalent, the number of correlation func-
tions is much smaller.
8.2.6 Short-Range Order (sro)
Configurations – Long-Range Order
(lro) Configurations
A configuration is called short-range or-
dered if the whole lattice constitutes one
class of lattice points (i.e., N
(1)
=N). As a
consequence, all the cluster correlation
functions are isotropic. A configuration is
called long-range orderedif at least two
classes of points are to be distinguished by
their different average occupation. These
classes of lattice points are usually called
sublattices. In Fig. 8-1, the face-centered
cubic unit cell is shown with a subdivision
of the lattice sites into four simple cubic
sublattices labeled 1 to 4 with N/4 points
each. With these sublattices, the super-
structures L1
2, L1
0and P4/mmm can be
described. Each point on a sublattice is sur-
rounded in its first-neighbor shell by four
points corresponding to each of the other
Figure 8-1.Subdivision of the face-centered cubic
(f.c.c.) unit cell into four simple cubic sublattices for
a characterization of the superstructures L1
2, L1
0and
P4/mmm. The occupations of the sublattices for
these structures are given in Table 8-1.www.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

581. sublattices. Therefore these structures are
called ordered in the first shell. The sublat-
tice occupation for these structures is given
in Table 8-1.
In the body-centered cubic structure,
four face-centered cubic sublattices with a
parameter a=2a
0can be defined as shown
in Fig. 8-2. With these sublattices, the
superstructures B2, D0
3, F4
–
3m, and B32
can be described. These structures are or-
dered in the first two neighbor shells. The
occupation of the sublattices is again given
in Table 8-1. The complete crystallo-
graphic specification of these superstruc-
tures is given in Sec. 8.7, Table 8-13.
We must therefore consider the follow-
ing variables: the points ·
s
1Ò, ·s
2Ò, ·s
3Ò,
·
s
4Ò; the pairs · s
1s
3Ò, ·s
1s
4Ò, ·s
2s
3Ò,
·
s
2s
4Ò, ·s
1s
2Ò, ·s
3s
4Ò; the triplets
·
s
1s
2s
3Ò, etc.; and, finally, the tetrahedron
·
s
1s
2s
3s
4Ò– in total, (2
4
– 1) = 15 vari-
ables. Of course, the variables have differ-
ent meanings in the two structures. In the
f.c.c. lattice, all the pairs are nearest neigh-
bors, while in the b.c.c. structure, the pairs
of type 1–2 and 3–4 are second-nearest-
neighbor pairs. A corresponding distinc-
tion among the triplets also has to be made.
Among these variables, we need four vari-
ables to define the isotropic f.c.c. sro state:
·
s
1Ò, ·s
1s
2Ò, ·s
1s
2s
3Ò, and · s
1s
2s
3s
4Ò;
one more variable, ·
s
1s
3Ò, is needed to
characterize the isotropic b.c.c. sro state.
The remaining variables characterize the
lro state. For many applications, it is suffi-
cient to identify the existence of lro by the
530 8 Atomic Ordering
Table 8-1.F.c.c. and b.c.c. superstructures and values of their lro parameters.
Designation/Spacegroup Point probabilities lro parameter Values for maximum
lro (binary)
A1 (Cu) A2 (Fe)
r
1
(i)= r
2
(i)= r
3
(i)= r
4
(i) x
1= x
2= x
3= 0 r
1
A= 1 – x
B
L1
0(CuAu) B2 (CsCl) r
1
(i)= r
2
(i)≠r
3
(i)= r
4
(i)x
1≠0, x
2= x
3= 0r
1
A= 1, r
3
A= 1 – 2x
B
L1
2(Cu
3Au) r
1
(i)≠r
2
(i)= r
3
(i)= r
4
(i)x
1≠x
2≠0, x
3= 0r
1
A= 1 – 4x
B, r
2
A= 1
D0
3(Fe
3Al)r
1
(i)≠r
2
(i)≠r
3
(i)= r
4
(i)x
1≠x
2≠0, x
3= 0r
1
A= 1 – 4x
B, r
2
A= 1,
x
BÙ0.25 r
3
A= 1
P4/mmm D0
3(Fe
3Al)r
1
(i)≠r
2
(i)≠r
3
(i)= r
4
(i)x
1≠x
2≠0, x
3= 0r
1
A= 0, r
2
A= 2 – 4x
B,
0.25 Ùx
BÙ0.5 0.25 Ùx
BÙ0.5 r
3
A= 1
B32 (NaTl)
r
1
(i)= r
3
(i)≠r
2
(i)= r
4
(i)x
1= 0, x
2≠x
3≠0 r
1
A= 1, r
2
A= 1 – 2x
B
F4
–
3m r
1
(i)≠r
2
(i)≠r
3
(i)≠r
4
(i) x
1≠x
2≠x
3≠0
0.25 Ùx
BÙ0.5
Figure 8-2.Subdivision of the body-centered cubic
(b.c.c.) unit cell into four f.c.c. sublattices with twice
the lattice constant, 2a
0, for a characterization of the
superstructures B2, D0
3and B32. The occupation of
the sublattices for these structures are given in Table
8-1. The (irregular) tetrahedron cluster 1234 used in
the CVM calculations is also shown.www.iran-mavad.com 
+ s e  l ⎨'4 , kp e  r  i ⎨&s ! 9 j+  N   0 e

582. 8.3 The Existence Domain and Configuration Polyhedron 531
difference between the point correlation
functions or linear combinations between
them (e.g., the same combinations as those
entering the structure factor for X-ray dif-
fraction). These linear combinations of the
point correlation functions are called lro
parameters. In the case of the b.c.c. struc-
ture, we usually define the lro parameters
x
1= ·s
1Ò+ ·s
2Ò– ·s
3Ò– ·s
4Ò
x
2= ·s
1Ò– ·s
2Ò
x
3= ·s
3Ò– ·s
4Ò
Table 8-1 lists the b.c.c. superstructures,
together with the values of the parameters.
If there is no long-range order, then all
the positions are equivalent and the point
probability takes one value 
r
n
(i)=x
i. Taking
In practice, the values of the pair correla-
tion functions for the sro states are ob-
tained from diffraction experiments. For
this purpose, we usually introduce the
Cowley–Warren sro parameters
a
ij
n–m
(Cowley, 1950), which are directly related
to the measured intensities. They are de-
fined as the deviation from the random
state:
The index n–mstands for the neighbor dis-
tance between the positions nand m. Tak-
ing again the 
t
ivalues as before (i.e., ±1
for the binary case and 1, 0, –1 for the ter-
nary case), we obtain the isotropic Cow-
ley–Warren sro parameters:
rrr
rr rr a
nm
ij
nm
ij
nm
ji
n
i
m
j
n
j
m
i
nm
ij
−
−
+−
=+
=[ ]( )
() () () ()
1
binary alloys: =
ternary alloys: =
=
AB AB
AB
AC AC
AC
a
ss s s
ss
a
ss ss ss
a
ss ss
a
nm
nm n m
nm
nm
nm nm nm
nm
nm nm
n
xx
xx
xx
xx
−
−
−
−
〈〉−〈〉〈〉
−〈 〉〈 〉
〈〉+〈〉+〈〉− −
−〈 〉+〈 〉+
1
241
4
4
4
22 22
22
()
mm
nm nm nm
xx
xx
BC CB
BC
=
−〈 〉−〈 〉+ 〈 〉− −
ss ss ss
222 2
241
4
()
the same 
t
ivalues as in the examples
treated previously in Sec. 8.2.3, we can 
use Eq. (8-6) and obtain the following 
values for the point correlation functions:
·
s
nÒ=·s
1Ò=x
A–x
Bin the case of a binary
alloy and ·
s
1Ò=x
A–x
C, ·s
1
2Ò=x
A+x
Cin
the case of a ternary alloy.
If there is not even any short-range or-
der, we consider the completely disordered
state. All the n-point correlation functions
then take a limiting value; for example, in 
a binary alloy, ·
s
1…s
rÒtakes the value
(x
A–x
B)
r
.
8.3 The Existence Domain
and Configuration Polyhedron
In the preceding section, we developed a
set of correlation functions (point, pair,
triplet, etc.) for characterizing atomic con-
figurations in multicomponent systems.
These correlation functions are indepen-
dent variables that define the configura-
tional space, but they can only vary within a
restricted range of numerical values due to
certain consistency conditions, which will
now be discussed. Outside this restrictedwww.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

583. range, the numerical values do not define
actually existing atomic configurations.
Therefore, this restricted range of values
has been called the existence domain, or
configuration polyhedron(Kudo and Kat-
sura, 1976). The dimension of the configu-
rational space depends either on the num-
ber of correlations which are taken into ac-
count or, equivalently, on the size of the
largest cluster. In the following sections the
existence domains will be investigated for
some simple examples. The method ap-
plied closely follows the procedure devel-
oped by Finel (1987). For the sake of sim-
plicity, the analysis will be restricted to bi-
nary alloys. For multicomponent systems,
the procedure is analogous but has to be
carried out on a computer.
Existence domains are most useful for an
analysis of ground states. This will be shown
in Sec. 8.4. The ground-state analysis will be
limited to a finite range of interactions (e.g.
pair interactions between first and second
nearest neighbors). Therefore, we are partic-
ularly interested in the part of the existence
domain that is the subspace of correlation
functions required for a treatment with fi-
nite-range interactions. This existence do-
main in the configurational subspace will
be investigated in detail in this section.
8.3.1 F.C.C. Structure,
First-Neighbor Interactions
The first step in this analysis is to make a
choice for the basic cluster. This choice
then determines the dimension of the con-
figurational space to be dealt with. It is 
natural to start with the simplest possible
basic cluster, the nearest-neighbor pair. The
configurational variables are then the point
correlation function x
1=·s
1Ò=x
A–x
B,
which defines the composition, and the
nearest-neighbor pair correlation function
x
2=·s
1s
2Ò=1–4r
12
AB(see Eq. (8-7)).
Adopting again the values 
t
i= ±1, which
were already used in the examples of Sec.
8.2, the variables can both vary in the inter-
val [–1, 1] (see Fig. 8-3): x
1takes the value
1 for pure A and –1 for pure B, x
2takes 1
for each pure component and –1 for an al-
loy composed only of A–B pairs. These
ranges, however, are not fully accessible,
for example because of the obvious con-
straints
r
12
AA70,r
12
BB70,r
12
AB70 (8-11)
The constraints in Eq. (8-11) yield the fol-
lowing inequalities:
r
12
AA70fi1+ 2x
1+ x
270
r
12
AB70fi1–  x
270 (8-12)
r
12
BB70fi1–2x
1+ x
270
532 8 Atomic Ordering
Figure 8-3.Projection of the configuration polyhe-
dron of a regular tetrahedron into the plane of corre-
lation functions x
1=·s
1Ò=x
A–x
Band x
2=·s
1s
2Òfor
different choices of clusters: (1) domain abg: nearest-
neighbor pair; (2) domain abfh: nearest-neighbor tri-
angle; (3) shaded area abcde: nearest-neighbor tetra-
hedron. The vertices a and b correspond to the pure
components A and B, c and e correspond to L1
2with
compositions B
3A and A
3B, and d to L1
0, composition
AB. The dashed line indicates the random configura-
tions. This line separates the existence domain into re-
gions of (ordered) configurations with preferential for-
mation of unlike pairs (lower part) and configurations
with preferential formation of like pairs (upper part).www.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

584. 8.3 The Existence Domain and Configuration Polyhedron 533
If the relations in Eq. (8-12) are taken as
equalities, they define a triangle in the
space (x
1,x
2), as shown in Fig. 8-3. Any
point inside the triangle abg fulfills these
relations. However, these relations are 
not sufficient to define the existence do-
main. In fact, the vertex g, for instance,
corresponds to 
r
12
AA=r
12
BB= 0 (introduce
x
1=·s
1Ò= 0 and x
2=·s
1s
2Ò= –1 into Eq.
(8-7)), which defines a configuration built
up with A–B pairs only. Such a struc-
ture cannot be formed in the f.c.c. lattice.
In this structure, the nearest-neighbor
bonds form a triangular network, and it is
not possible to form a triangle with only
A–B pairs. Point g in Fig. 8-3 thus corre-
sponds to a physically impossible state.
Obviously, there must be further restricting
relations.
The conditions 
r
ij
12
&1 do not introduce
new constraints because they are automati-
cally fulfilled together with the inequalities
in Eq. (8-11). For the present choice of 
t
i
values, this is immediately seen from the
definition in Eq. (8-7). The general case
follows from Eq. (8-2), which yields the re-
lation:
r
12
AA+ r
12
BB+ r
12
AB+ r
12
BA= 1 (8-13)
If the constraints in Eq. (8-11) are fulfilled,
then Eq. (8-13) is a sum of positive terms
which can only be equal to one if each term
is less than or equal to one.
Hence, we have to deduce further in-
equalities from higher-order clusters. Let
us consider the nearest-neighbor triangle
123 as the next cluster, see Fig. 8-1. The
new variable x
3=·s
1s
2s
3Òmust be intro-
duced, and the configurational space is
now three-dimensional. The constraints for
the triplet probabilities are
(8-14)
r
123
AAA70fi
r
123
AAB70fi
r
123
ABB70fi
r
123
BBB70fi
Because we are only interested in the do-
main spanned in the subspace (x
1,x
2), we
have to eliminate the variable x
3. In order
to converse the positive value of all the ex-
pressions, this elimination can only be per-
formed by additive combinations of the in-
equalities. This yields the three expressions
in Eq. (8-12) already obtained from the
pairs, plus one more inequality:
1+ 3x
270
Taken as an equality, this equation defines
the line f–h in Fig. 8-3, and we have to ver-
ify again if the new vertices f and h corre-
spond to physically accessible states. Point
f corresponds to a state with no triangle of
type AAA, AAB, BBB (i.e., a state built up
only with triangles of type ABB). It is easy
to see that this is impossible in the f.c.c.
structure. Thus this procedure has to be
continued in order to find further restrict-
ing conditions.
The next higher cluster that may be con-
sidered is the nearest-neighbor tetrahedron
r= 1234, see Fig. 8-1. In this case, the new
variable x
4=·s
1s
2s
3s
4Òneeds to be intro-
duced. The tetrahedron probabilities result
in the following consistency relations:
1331
1111
1111
1331
1
0
1
2
3−−
−−
−−
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
x
x
x
7
14641
12021
10201
12021
14641
1
0
1
2
3
4−−
−
−−
−−
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
x
x
x
x
7 (8-15)
r
1234
AAAA70fi
r
1234
AAAB70fi
r
1234
AABB70fi
r
1234
ABBB70fi
r
1234
BBBB70fiwww.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

585. Taken as equalities, these equations define
a closed polyhedron in the space (x
1,…,x
4).
Because we are still interested in the 
subspace (x
1,x
2) we have to eliminate x
3
and x
4. This is done by taking suitable ad-
ditive combinations of the inequalities in
Eq. (8-15). The result is the polygon abcde
in Fig. 8-3, which differs from the pre-
vious one in that the vertices f and h 
are removed, and the vertices c, d, e are
added. It is easy to see that all these ver-
tices correspond to possible atomic con-
figurations in the f.c.c. structure; c and e
correspond to the L1
2structure with stoi-
chiometric compositions B
3A and A
3B, re-
spectively, and d corresponds to the L1
0
structure with composition AB. The exis-
tence domain is given by the shaded area in
Fig. 8-3.
The regular tetrahedron is thus the small-
est basic cluster that allows us to define the
existence domain in the configuration
space of an f.c.c. lattice with nearest-neigh-
bor interactions. Consequently, this tetra-
hedron is also the smallest basic cluster
upon which a statistical treatment should
be based. It is possible, of course, to take a
higher cluster containing the tetrahedron as
a subcluster. In such a case, the dimension
of the configurational space is increased
but the projection of the domain into the
subspace of point correlations and nearest-
neighbor pair correlations remains the
same. If we had considered a basic cluster
that does not contain the tetrahedron, we
would have obtained a polygon with some
vertices, which again would not corre-
spond to accessible states. The procedure
would have to be continued until a cluster
containing the tetrahedron were finally se-
lected.
The procedure of obtaining the vertices
of the configuration polyhedron outlined
above becomes more and more inconven-
ient with increasing cluster size, i.e., with
an increasing number of correlation func-
tions and, consequently, with more in-
equalities that have to be taken into ac-
count in order to define the existence do-
main. Therefore we look for a simplifica-
tion. At first we notice that each one of the
consistency relations, when taken as an
equality, defines a face of a polyhedron. 
In the cases studied before (see, for exam-
ple, Eq. (8-14) or (8-15)), the number of
consistency relations was one more than
the dimension of the configurational space.
The polyhedron was thus built by d+1
faces. Therefore, the polyhedron is a sim-
plex, which is necessarily closed and con-
vex. Generally, if dis the dimension of 
the space, a simplexis a polyhedron with
d+ 1 faces. The intersection of dfaces 
defines a vertex, a simplex thus has d–1
vertices. The coordinates of the vertices 
are obtained by solving the systems of d
equations which can be chosen out of the
whole set of d+ 1. The simplification
comes from a very convenient way of ob-
taining these coordinates: Finel (1987)
showed that the values obtained from 
these solutions are the same as those ob-
tained simply by dividing the elements in
each column of the matrix by the cor-
responding values of the first row. The 
coordinates of the vertices are the elements
in each row following the first column,
which always contains unity and does not
define a coordinate. Performing this proce-
dure with the matrix of Eq. (8-14), we 
directly obtain the coordinates given in 
Table 8-2. Taking the projection into the
subspace (x
1,x
2) means we conserve only
the first two coordinates. This exactly de-
fines the polygon abfh, which was found
before by means of the elimination proce-
dure.
Applying the same procedure to Eq. 
(8-15) immediately yields the coordinates
of the configuration polyhedron in the
534 8 Atomic Orderingwww.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

586. 8.3 The Existence Domain and Configuration Polyhedron 535
four-dimensional space (x
1,…,x
4), see 
Table 8-3. The vertices abcde in Fig. 8-3 of
the two-dimensional polygon in the sub-
space (x
1,x
2) can immediately be taken
from Table 8-3.
8.3.2 F.C.C. Structure, First and
Second Neighbor Interactions
First and second neighbor interactions
correspond more closely to the situation
encountered in real alloys (Inden, 1977a)
than the previous case does. The internal
energy contains one more term, the second-
nearest neighbor interaction. Consequently
we will finally be interested in the exis-
tence domain within the subspace of point
and first and second neighbor pair correla-
tion functions. The interesting part of the
existence domain will be a polyhedron in
this three-dimensional space.
The analysis begins with the octahedron
r= 234567 in Fig. 8-4, which is defined as
a basic cluster because it includes both
types of interaction. For this cluster, nine
correlation functions have to be intro-
duced:
x
1= ·s
2Ò x
6= ·s
2s
3s
4s
7Ò
x
2= ·s
2s
3Ò x
7= ·s
3s
4s
5s
6Ò
x
3= ·s
2s
7Ò x
8= ·s
2s
3s
4s
5s
7Ò
x
4= ·s
2s
3s
4Ò x
9= ·s
2s
3s
4s
5s
6s
7Ò
x
5= ·s
2s
3s
7Ò
x
1again corresponds to x
A–x
B. The exis-
tence domain will be a polyhedron in 9-
dim. space (x
1,…,x
9), which is defined by
the consistency relations for the various
configurations on the octahedron cluster:
Table 8-3.Coordinates of the vertices of the config-
uration polyhedron shown in Fig. 8-3.
Configuration Correlation functions
Ve r t e xx
1 x
2 x
3 x
4
Pure A a 1111
L1
2(A
3B) e 1/2 0 –1/2 –1
L1
0(AB) d 0 –1/3 0 1
L1
2(AB
3) c –1/2 0 1/2 –1
Pure B b –1 1 –1 1
ij k
ij k
…
−−−−
−−−
−− −
−−
−−−
…
…
AAAAAA
BAAAAA
BBAAAA
BAAAAB
BBBAAA
BBAAAB
BBBAAB
ABBBBA
BBBBAB
BBBBBB
:
:
:
:
:
:
:
:
:
:
=
r
23 7 6
1
2
1 6 12 3 8 12 12 3 6 1
14410 04141
12010 40121
12438 44321
10030 00301
10410 04101
1−−− −− −
−− −−
−−−−
−−− −
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
2438 44321
12010 40121
14410 04141
1 6 12 3 8 12 12 3 6 1
1
1
2
3
4
5
6
7
8
9x
x
x
x
x
x
x
x
x
⎟⎟
⎟
⎟
⎟
⎟
⎟
⎟
70
Table 8-2.Coordinates of the vertices shown in Fig.
8-3.
Vertex x
1 x
2 x
3
a111
h 1/3 –1/3 –1
f –1/3 –1/3 1
b–11–1www.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

587. Taken as equalities, these 10 equations de-
fine a simplex in the 9-dim. configurational
space. From these equations, the vertices of
the simplex can be calculated either by
solving nine selected equations out of the
whole set of 10 or obtained directly by
Finel’s method outlined above.
Let us consider the projection of this
simplex into the subspace (x
1,x
2,x
3). This
projection is obtained from columns 2–4
of the matrix. The result is shown in Fig. 
8-5. A verification of whether or not all the
vertices correspond to possible atomic ar-
rangements is necessary. For example, ver-
tex (1/3, –1/3, 1) does not correspond to an
existing state because it would be built
only with octahedra of type BAAAAB.
This is topologically impossible in the
f.c.c. structure. Therefore, we have to 
consider further constraints, for instance
those resulting from the nearest-neighbor
tetrahedron, analyzed before. We can thus
use the previous results. In the space
(x
1,x
2,x
3), the previously obtained exis-
tence domain would be the rectangular
prism built on the basis given by the shaded
area in Fig. 8-3 (see also Fig. 8-6). If we
now take into account the constraints im-
posed by both the tetrahedron and the octa-
536 8 Atomic Ordering
Figure 8-4.Tetrahedron and octahedron clusters in
the face-centered-cubic unit cell.
Figure 8-5.Projection of the configuration polyhe-
dron of an octahedron cluster r= 234567 (see Fig. 8-
4) into the subspace of point and first and second
neighbor pair correlation functions: x
1=·s
2Ò,
x
2=·s
2s
3Ò, x
3=·s
2s
7Ò, respectively. Not all the ver-
tices correspond to existing states. The shaded area
represents the section for x
3= 0, i.e. plane (x
1,x
2).
Comparison with Fig. 8-3 reveals that this section de-
fines a larger domain. This shows that the octahedron
does not impose the same constraints as the tetrahe-
dron cluster.
Figure 8-6.Configuration polyhedron of a nearest-
neighbor tetrahedron cluster 1234 (Fig. 8-4) in the space (x
1,x
2,x
3). This cluster does not contain sec-
ond-nearest neighbor pairs. The polyhedron is thus a rectangular prism built with the existence domain in Fig. 8-3 as basis.www.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

588. 8.3 The Existence Domain and Configuration Polyhedron 537
hedron, we will have to intersect the two
polyhedra. The result of the intersection of
Fig. 8-5 and Fig. 8-6 is shown in Fig. 8-7,
and the coordinates of the vertices are
given in Table 8-4. It is easy to show that
all the vertices of this polyhedron corre-
spond to existing atomic configurations.
Consequently, in the f.c.c. structure with
first and second nearest-neighbor interac-
tions, we have to introduce at least the reg-
ular tetrahedron and the octahedron as ba-
sic clusters. Fig. 8-7 also illustrates the
statement made in the previous paragraph
that if we reduce the interaction range to
nearest neighbors and keep the two basic
clusters, the projected existence domain in
the space (x
1,x
2) does not change due to
the introduction of the octahedron, a larger
cluster than the tetrahedron.
8.3.3 B.C.C. Structure, First and
Second Neighbor Interactions
The treatment of the b.c.c. structure is
much simpler than that of the f.c.c. struc-
ture because an irregular tetrahedron as 
a basic cluster already contains first and 
second neighbor distances, as shown in
Fig. 8-2. With this cluster, we must in-
troduce the variable x
1=·s
1Ò, x
2=·s
1s
3Ò,
x
3=·s
1s
2Ò, x
4=·s
1s
2s
3Ò, x
5=·s
1s
2s
3s
4Ò.
This consistency relations are
Figure 8-7.Intersection of the two configuration
polyhedra corresponding to the tetrahedron and the
octahedron, Fig. 8-4 and Fig. 8-5. All vertices of this
polyhedron correspond to existing configurations on
the f.c.c. lattice. The corresponding superstructures
are indicated. Notice the symmetry with respect to
the stoichiometric composition AB. The vertical sec-
tions represent the existence domains for a fixed al-
loy composition, here AB and A
3B.
Table 8-4.Coordinates of the vertices of the config-
uration polyhedron shown in Fig. 8-7.
Configuration Correlation functions
x
1 x
2 x
3
Pure A (A1) 1 1 1
A
5B, AB
5 ±2/3 1/3 1/3
A
3B, AB
3(L1
2)±1/2 0 1
A
3B, AB
3(D0
22)±1/2 0 2/3
A
2B, AB
2(Pt
2Mo) ±1/3 –1/9 1/9
A
2B, AB
2 ±1/3 0 –1/3
AB (L1
0)0–1/31
AB (L1
1)00–1
A
2B
2 0–1/31/3
Pure B (A1) –1 1 1
:
:
:
:
:
:
ijkl
x
x
x
x
x
ijkl
AAAA
BAAA
BABA
BBAA
BBBA
BBBB
=
r
1234 4
1
2
3
4
5
1
2
144241
120021
100201
104201
120021
144241
1
0
−−
−
−
−−
−−
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
7www.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

589. Taken as equalities, these equations define
a simplex in the 5-dim. configurational
space. All the vertices correspond to exist-
ing atomic configurations. Its projection
into the subspace (x
1,x
2,x
3) defines the
configuration polyhedron shown in Fig. 
8-8. The coordinates of the vertices are
given in Table 8-5.
8.4 Ground States
The description of the configurations
will now be used to derive the stability of
the configurations at T= 0 K. The most
stable state, the ground state, is given by
the minimum of the (grand canonical)
internal energy U
˜
(V,T,
m
A,m
B, …) for
given values T,V,
m
iobtained by variation
of the internal configurational variables,
i.e. point, pair, triplet, etc. correlation func-
tions. In practice it is preferred to consider
the (canonical) internal energy U(T,V,N
A,
N
B, …) with the particle numbers N
ias
variables. The grand canonical energy is
then obtained by the Legendre transforma-
tion of U:
The general expression for U
˜
can be
written
(8-16)
The summations have to be taken over all
equivalent points, pairs, etc. In this formu-
lation the higher-order cluster interactions
(beyond pairs) are not the total energies of
these clusters, but correction terms. De-
pending on the structure, the clusters of a
given class may share subclusters. This
sharing of subclusters has to be taken into
account if total energies are used.
8.4.1 Pair Interactions
The concept of pair interactions has a
long tradition and has been supported by
the theoretical work of Gautier (1984), 
Bieber and Gautier (1984a, b, 1986, 1987),
and Turchi et al. (1991a, b), who came to
the conclusion that pair interactions play a
˜
(),
() ( )
(),,
()
() () ()
UppV p
ppV p
nm i j
n
i
m
j
ij
nm
nmq i j k
n
i
m
j
q
k
ijk
nmq
ni
n
i
i
=∑∑ ∑ ∑
∑∑ +
×+…+
m
˜
UU N
i
K
ii
=
=
+∑
1
m
538 8 Atomic Ordering
Figure 8-8.Projection of the configuration polyhe-
dron corresponding to the irregular tetrahedron
r= 1234 of the b.c.c. lattice (see Fig. 8-2) into the
subspace of point and first and second neighbor pair
correlation functions x
1=·s
1Ò=x
A–x
B, x
2=·s
1s
3Ò,
x
3=·s
1s
2Ò, respectively. All vertices correspond to
existing ordered atomic configurations.
Table 8-5.Coordinates of the vertices of the config-
uration polyhedron shown in Fig. 8-8.
Configuration Correlation functions
x
1 x
2 x
3 x
4 x
5
Pure A (A2) 1 1111
A
3B, AB
3(D0
3)±1/2 0 0 ≠1/2 –1
AB (B2) 0 –1 1 0 1
AB (B32) 0 0 –1 0 1Pure B (A2) –1 1 1 –1 1www.iran-mavad.com 
+ s e  l ⎨'4 , kp e  r  i ⎨&s ! 9 j+  N   0 e

590. 8.4 Ground States 539
dominant role in alloys, while higher-order
cluster interactions are of minor impor-
tance. This appears to be a good approxi-
mation for the class of alloys with negli-
gible relaxation effects. Furthermore, pair
correlations can be determined by diffrac-
tion experiments (see, e.g., Schönfeld,
1999).
Grouping together all the energy contri-
butions from positions nmwith the same
distance k, we obtain the following results
for the internal energy:
where 
n
kdefines a point in the k-th neigh-
bor shell of an arbitrary position in sublat-
tice 
ntaken as the origin, and z
n
(k)
is the cor-
responding coordination number. We ob-
tain
and
where position 0 stands for any position in
the crystal taken as the origin. Therefore,
the energy is
(8-17)
This expression contains only the isotropic
pair probabilities because we assumed 
isotropy for the pair interactions. Conse-
quently, it is not possible to distinguish lro
˜ ()() ()
UN zV N
kij
k
ij
k
k
ij
i
i
i
=
1
2
0 0∑∑ ∑ +rmr
n
n
n
rr
=
=
1
0
L
i i
NN∑
() ()
n
n
n
nn
rr
=
=
1
0
L
k ij k
k
ij
zN zN
k∑
() () ()
˜ () ()
,
()() ()
() ()
,
() () ()
(
UzNppV
Np
VzN
N
L
k
k
ij
ij
ij
k
i
i
i
kij
ij
k
L
k ij
i
i
L
k
k
=
=
=
=
=n
n
nn
n
n
n
n
n
nn
n
n
m
r
m
1
1
1
1
2
1
2
∑∑ ∑
∑
∑∑ ∑
∑∑
⎧
⎨
⎩
〈〉
+〈〉
⎫
⎬
⎭
+
))()
r
n
i
from sro by energetic arguments. In fact,
lro results from a topological constraint.
Beyond a critical number of unlike bonds,
it is no longer possible to arrange these
bonds on a lattice with the constraint to
maintain the equivalence of all lattice
points.
In Eq. (8-17), no reference state is speci-
fied. We will refer the energy to the me-
chanical mixture of the pure components 
in the same crystal structure as the alloys
and obtain the following expression from
Eq. (8-17):
where D
m
i=m
i–m
i
0and W
(k)
=–2V
(k)
AB
+
V
(k)
AA
+V
(k)
BB
. The W
(k)
are called exchange-
energyparameters. They take positive val-
ues for an ordering tendency in the k-th
shell and negative values for a separation
tendency. In order/disorder problems, we
are not usually interested in the size of the
system, and the quantity DU
˜
/Ncan then be
treated. If we fix temperature and pressure
and consider only equilibrium states, the
number of independent chemical potentials
is reduced by one. For convenience we often
take D
m
A+Dm
B= 0 and define an effective
chemical potential 
m*=–
1
2
(Dm
A–Dm
B).
Finding the ground states is equivalent
to finding the minimum of E
˜
, which is de-
fined as the part of the internal energy that
depends only on the configurational vari-
ables:
˜
˜ () ()
E
U
N
zW x
k
kk
s
k
ss
= = (8-19)
=
D
+
∑∑
+
1
8
1
1
F
D
DD DD
˜˜
()
()()
()
()
() () () ()
UU N z
xV xV N
NN
N
zW
N
zW
k
k
kk
i
i
i
k
kk
k
kk
k
= (8-18)
=
AAA
()
BBB
()
=A,B
AB AB
−
×+−
++ −〈〉
−+〈〉∑
∑
∑∑
1
2
28
88
0
0
0
0
mr
mm mms
sswww.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

591. with x
s=·s
0Ò, ·s
0s
1Ò,…,·s
0s
kÒand 
8.4.1.1 Ground State Energies
Eq. (8-19) shows how the energy E
˜
de-
pends on the configurational variables x
s,
that is, on the composition by x
1=·s
0Ò=
x
A–x
B, and on the pair correlation func-
tions by x
2,x
3, … . Among the variety of
possible states, there are also those with the
lowest energy E
˜
min. These states will now
be determined using a geometrical inter-
pretation of Eq. (8-19).
Firstly, Eq. (8-19) is a linear relation. This
implies that all those values x
swhich fulfill
Eq. (8-19) define a (hyper)plane in the con-
figurational space. It will now be shown
that the energy E
˜
is proportional to the dis-
tance of this (hyper)plane from the origin.
To demonstrate this, we introduce the
vector x=(x
1,x
2,…,x
k–1) and a unit vec-
tor n=
F/|F|=|F|
–1
(m*, z
(1)
W
(1)
/8, …,
z
(k)
W
(k)
/8). Eq. (8-19) can then be writen
in vector form:
(8-20)
where dis the distance of the hyperplane
from the origin. This distance takes posi-
tive or negative values depending on the
directions of the two vectors nand x. The
factor |
F|depends only on the chemical
potential, the exchange energies and the
coordination numbers, that is, it defines the
alloy under consideration. If we change the
chemical potential (which controls the
composition) and/or the energy parame-
ters, then the direction of nchanges (ex-
cept if we change all the components 
F
iby
a common factor), as does the value of d.
If, on the other hand, we consider a
given alloy, which means we keep vector 
F
fixed but vary the values of x, then the nor-
mal vector ndoes not change. This means
()
˜
xn|
||
==
E
d
FF
F
s
kk zW zW=m*, , , .
() () ( ) ( )1
8
1
8
11
…
that in this instance the planes of constant E
˜
are parallel to each other, and it follows
immediately from Eq. (8-20) that the ex- trema of E
˜
must correspond to extreme val-
ues of x. Because the origin is inside the
existence domain for the present choice of
t
i= ±1, the value E
˜
minmust be negative.
Searching the ground state thus becomes equivalent to finding, for a given alloy (i.e., for a given n=
F/|F|), the values of x
that yield the most negative value for the distance dfrom the origin. If we recall that
the existence domain is a closed polyhe- dron, it follows that the ground states must correspond to vertices of the polyhedron. This will now be illustrated with some ex- amples that have already been considered in previous sections.
8.4.1.2 F.C.C. Structure,
First-Neighbor Interactions
With first-neighbor interactions in the
f.c.c. structure, the energy E
˜
depends only on
the compositional parameter x
1=x
A–x
B,
and on one configurational parameter, the
correlation function of nearest-neighbor
pairs x
2=·s
0s
1Ò=1–4r
01
AB(see Eq. (8-
7)), where position 1 is a nearest-neighbor
site to a central site 0. Therefore, the exis-
tence domain in Fig. 8-3 is relevant to this
case. It follows from the foregoing discus-
sion that the ground states are given by the
vertices in Fig. 8-3 (i.e., pure A, pure B,
L1
2with compositions A
3B and AB
3, and
L1
0with composition AB). Fig. 8-9 illus-
trates this point with some examples.
If, for example, 
F
1=m*/|F|= 0, then the
vector 
F=(0,F
2) and, consequently, nis
parallel or antiparallel to the x
2-axis, de-
pending on the sign of 
F
2=––
12
8
W
(1)
. The
lines E
˜
= const. are then parallel to the x
1-
axis. This can also be seen from Eq. (8-20),
which reduces to
F
2x
2= E
˜
540 8 Atomic Orderingwww.iran-mavad.com 
+ s e  l ⎨'4 , kp e  r  i ⎨&s ! 9 j+  N   0 e

592. 8.4 Ground States 541
These lines are shown in Fig. 8-9. If we
consider an alloy with ordering tendency
(W
(1)
>0¤F
2> 0), the value E
˜
minis
reached for the most negative x
2value,
which is x
2= –1/3. This point corresponds
to the L1
0structure AB. If we consider an
alloy with separation tendency (W
(1)
<0
¤
F
2< 0), the value E
˜
minis reached for the
most positive x
2value, which is x
2=1. In
this case, the line E
˜
=E
˜
mincoincides with a
side of the configuration polygon. This line
joins the vertices corresponding to the pure
components A and B. The ground state is
degenerate: it could be pure A and pure B
in any proportion juxtaposed to each other,
or any other arrangement of A and B
which, in the limit of NÆ∞, does not 
form A–B bonds (e.g., constant number of
2-dim. boundaries).
Next, we may consider the case 
F
2=––
12
8
W
(1)
= 0, which corresponds to an
ideal solution. The lines E
˜
= const. are par-
allel to the x
2-axis in this case, because nis
parallel to the x
1-axis. Eq. (8-20) now re-
duces to
F
1x
1= E
˜
For a positive value of 
F
1=m*=
–
1
2
(Dm
A–Dm
B), the most negative value 
of E
˜
is obtained for x
1=(x
A–x
B) = –1, i.e.,
corresponding to pure B, and a negative
value of 
F
1corresponds to pure A.
The procedure outlined above can be re-
peated for any direction of the vector
F,
see Fig. 8-9. As long as the lines of con-
stant E
˜
pass through a vertex of the poly-
gon, a unique ground state can be defined.
If these lines coincide with the sides of the
polygon, the ground state is degenerate. It
could then be any mixture of the two states
corresponding to the two vertices defining
the side of the polygon. It was mentioned
Figure 8-9.Lines of constant grand canonical energy E
˜
(Eq. (8-20)) for given sets of values  F
1=m*=
–
1
2
(m
A–m
B) and F
2=–
3
4
W
(1)
in the case of the f.c.c. lattice and nearest-neighbor interactions. The configuration
polygon is that of Fig. 8-3. Extrema of the energy correspond to lines passing through the vertices of the poly-
gon. Depending on the sign of 
F
1and F
2the line through a vertex corresponds to a minimum or a maximum.
The vertices thus define the ground states. For particular 
F
1, F
2values the lines coincide with the sides of the
polygon. For these values, the ground state is degenerate.www.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

593. before that the proportion of the two states
differing in composition is not defined 
because in these cases of degeneracy the
overall composition of the system is not
defined by a fixed value of the chemical
potential. We can say, however, that the
state is two-phase.
Finally, we can determine a so-called
ground-state diagram, which is shown in
Fig. 8-10. It is worth noting that the single-
phase fields are sectors in the (
F
1,F
2)
plane limited by lines originating from the
origin. It is thus sufficient to know the di-
rections of those lines. Furthermore, the di-
agram is symmetric with respect to 
F
1=0.
When higher-order interactions are in-
volved, as in the next paragraph, the deter-
mination of the ground-state diagram is not
that simple, and it is useful to have a more
analytical method for its determination.
Such a method will now be introduced and,
for the purpose of illustration, will be ap-
plied to the present very simple case.
The ground-state diagram displays the
ranges of 
F
1and F
2values for which a
given state, e.g., L1
0, is stable. We thus
interpret Eq. (8-19) inversely to that of Eq.
(8-20): given a vertex in the configura-
tional space, e.g., vertex x
v= (–1 3, 0) cor-
responding to L1
0, we define a unit vector
n*=x
v/|x
v|. Eq. (8-19) can then be rewrit-
ten as
(
F|n*) = E
˜
/|x
v|= d* (8-21)
In a method similar to that used with Eq.
(8-20), we interpret Eq. (8-21) as the equa-
tion of a line in the space (
F
1,F
2) with a
distance d*=E
˜
/|x
v|from the origin. If we
fix the value of E
˜
, then all vectors
Fful-
filling Eq. (8-21) must end in the line with
the normal n* and with distance d*. In Fig.
8-10, the result is shown for E
˜
= –1 (energy
units). It shows that the vectors fulfilling
Eq. (8-21) describe a polygon. The coordi-
nates of the vertices of this polygon are
given in Table 8-6. This polygon is the dual
polygon to the configuration polygon 
(Finel, 1987), and the space (
F
1,F
2) is
called the dual space. A face of the dual
polygon corresponds to each vertex of the
configuration polygon in the direct space
(x
1,x
2). All vectorsFpointing to the same
face of the polygon define alloys that ex-
hibit the same ground state given by the as-
sociated vertex of the configuration poly-
gon. If we vary E
˜
, then the polygon shrinks
or blows up, respectively, and the vertices
of the dual polygon vary along the lines
that define the two-phase states in Fig. 8-
10. It is easy to verify that these lines are
542 8 Atomic Ordering
Figure 8-10.Ground-state diagram and dual poly-
gon (to Fig. 8-9) for the case of an f.c.c. lattice with
nearest-neighbor interactions. The axes are
F
1=m*=–
1
2
(m
A–m
B) and F
2=–
3
4
W
(1)
. The solid lines
are radius vectors which subdivide the plane into re-
gions with a defined ground state. The directions of
these vectors are defined by the vertices of the dual
polygon to Fig. 8-3. It corresponds to a fixed value of
the energy, here E
˜
= –1 (energy units), and is ob-
tained from Eq. (8-21). The dual polygon already
contains complete information on the ground-state
diagram. For 
F
1, F
2values corresponding to these
lines, the ground state is degenerate.www.iran-mavad.com 
+ s e  l '4 , kp e  r  i &s ! 9 j+  N   0 e

594. 8.4 Ground States 543
vertical to the faces of the configuration
polygon in Fig. 8-3.
The generalization of these results to
higher dimensions is straightforward. A dual
polyhedron defined by Eq. (8-21) is asso-
ciated with the (direct) configuration poly-
hedron in such a way that each vertex of the
direct polyhedron corresponds to a face of the
dual polyhedron, and vice versa. In order to
get the vertices of the dual polyhedron, we
have to solve Eq. (8-21) simultaneously for
the number of vertices of the direct polyhe-
dron required to define the (hyper)plane.
The application of this in three dimensions
will be shown in the next two paragraphs.
8.4.1.3 F.C.C. Structure, First and
Second Neighbor Interactions
With first and second neighbor interac-
tions in the f.c.c. structure, the energy E
˜
depends on one more configurational pa-
rameter, the second neighbor pair corre-
lation function x
3=·s
0s
2Ò=1–4r
02
AB(see
Eq. (8-7)). The relevant existence domain
is now that shown in Fig. 8-7. We can again
proceed as previously.
Let us first consider 
F
3=––
12
8
W
(2)
=0.
The normal vector ncorresponding to the
planes of constant energy E
˜
is perpendicu-
lar to the axis x
3in this case. If we consider
an alloy with an ordering tendency in the
first shell, 
F
2=––
12
8
W
(1)
> 0, then nis
oriented in the half space x
2> 0, and the
planes corresponding to E
˜
minare those
passing through the vertices pure A, L1
2
with compositions A
3B or AB
3, L1
0with
composition AB, and pure B, depending on
the value of x
1=m*=–
1
2
(Dm
A–Dm
B). This
is in accordance with the results already
obtained in Sec. 8.4.3 (see Fig. 8-10).
However, an additional result can be de-
duced from Fig. 8-7: the energy planes pass
simultaneously through the vertices corre-
sponding to the structures D0
22and L1
2
(both either A
3B or AB
3), as well as L1
0
and A
2B
2. The ground states are thus de-
generate in these instances.
The complete ground-state diagram is
obtained from the determination of the dual
polyhedron associated with the existence
domain of Fig. 8-7. This dual polyhedron is
shown in Fig. 8-11. All
Fvectors pointing
to (or piercing) a given face of the dual
Figure 8-11.Dual polyhedron (to Fig. 8-7) for the case of an f.c.c. lattice with first and second neighbor inter-
actions. The axes are 
F
1=m*=–
1
2
(m
A–m
B), F
2=–
3
4
W
(1)
, and F
3=–
3
4
W
(2)
. The dual polyhedron defines the
ground-state diagram and corresponds to a constant energy E
˜
= –1 (energy units). All radius vectors pointing to
the same face correspond to the same ground state, which is labeled for each face, except the top, which is L1
1.
Vectors coinciding with the boundaries of the faces correspond to degenerate ground states.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

595. polyhedron define alloys with the same
ground state, which is defined by the corre-
sponding vertex of the configuration poly-
hedron. The ground-state diagram thus di-
vides the dual space into (hyper)cones
formed by the vectors from the origin to the
borderlines of the faces. Each cone defines
a ground state. Due to graphical limita-
tions, those cones are not shown in Fig. 
8-11, but the ground states are labeled 
at each face of the dual polyhedron. The
coordinates of the dual vertices are given in
Table 8-6. They have been derived by 
inserting the vertices in Table 8-3 into Eq.
(8-21) for E
˜
min=–1.
8.4.1.4 B.C.C. Structure, First and
Second Neighbor Interactions
In the first and second neighbor interac-
tions of the b.c.c. structure, the energy E
˜
depends on the same variables (x
1,x
2,x
3)
as in the preceding example, and the rele-
vant existence domain is that shown in Fig.
8-8. We can now construct the dual polyhe-
dron, solving Eq. (8-21) for E
˜
= –1 and us-
ing the vertex coodinates in Table 8-4. The
coordinates of the dual vertices are given in
Table 8-6, and the dual polyhedron is
shown in Fig. 8-12.
8.4.1.5 Energy Minimum at Constant
Composition
From a metallurgical point of view it
seems more natural to consider the com-
position of a system an independent vari-
able which we can control, rather than the
544 8 Atomic Ordering
Table 8-6.Coordinates of the dual polyhedra shown
in Figs. 8-10, 8-11 and 8-12.
Tetra- Tetrahedron– Tetrahedron
hedron Octahedron b.c.c.
f.c.c. f.c.c.
F
1F
2 F
1F
2F
3F
1F
2F
3
21 ±210 00–1
23 ±230 ±210
–2 3 ±18/7 12/7 3/7 ±2 2 1
–2 1 ±2 0 1 ±2 0 1
0–1 ±2 4 1 0–21
0–21
00–1
Figure 8-12.Dual polyhedron (to Fig. 8-8) for the case of an b.c.c. lattice with first and second neighbor inter-
actions. The axes are 
F
1=m*=–
1
2
(m
A–m
B), F
2=W
(1)
, and F
3=–
3
4
W
(2)
. The dual polyhedron defines the
ground-state diagram and corresponds to a constant energy E
˜
= –1 (energy units). All radius vectors pointing to
the same face of the polyhedron correspond to the same ground state, which is labeled for each face, except the top one, which is B32.www.iran-mavad.com 
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596. 8.4 Ground States 545
chemical potential. However, this is not
possible. We can, of course, control the
overall composition of our system by tak-
ing the appropriate amounts of the pure
substances, closing the system against ma-
terials exchange, and producing a certain
configuration (e.g., by heat treatment). If
we thereby equilibrate the system at a
given temperature, the system will attain
its energy minimum by internal reactions.
These reactions can be the formation of or-
dered atomic arrangements (homogeneous
single-phase state) or the formation of het-
erogeneous microstructures. In the latter
case, we have to introduce new variables in
addition to the configurational variables,
for example, the volume of produced
phases, describing the advance of these re-
actions. At equilibrium, the driving force
for the reactions must be zero, that is, the
chemical potentials of all the reaction prod-
ucts must be the same. Because the overall
composition is fixed, the amount of the
phases present is also fixed. This is differ-
ent from a grand canonical treatment. The
energy in this case is given by the canoni-
cal energy E , which is given by the expres-
sion
(8-22)
This equation is similar to Eq. (8-19), ex-
cept that the number of variables is reduced
by one. The geometrical considerations can
be applied as before. In Fig. 8-7, two sec-
tions corresponding to the compositions
x
1=x
A–x
B= 0 and 0.5 (AB and A
3B) are
shown as shaded areas. Since in this in-
stance we are confined to a planar section,
the loci of constant energy are lines, and
the minimum energy corresponds to the
lines passing through the vertices of the
polygon. We will consider the case x
1= 0.5
in detail, and we thus have to consider the
polygon SLDMK in Fig. 8-7.
EEx x
s
k
ss
==
=
˜
−
+
∑11
2
1FF
The states corresponding to L and D
(L1
2and D0
22) are ground states because
they are vertices of the configuration poly- hedron. S is not a vertex in this figure, but lies on the line joining the states pure A and pure B. This line corresponds to con- figurations that do not contain any A–B pairs. There is no other configuration pos- sible except for pure A and pure B taken to- gether in the proportion A
3B. Of course,
the planar interface, which is necessarily formed, contains A–B pairs. In the thermo- dynamic limit of NÆ∞, however, the rela-
tive number of these pairs approaches zero and can be neglected.
The point K lies on the line joining the
vertices pure A and L1
1(AB). Here again
the two limiting phases can be taken to- gether in the right proportion to build up a two-phase configuration at point K. The possibility of another homogeneous ar- rangement needs to be verified, though. But K is not a vertex of the configuration poly- hedron in Fig. 8-7. Hence, an additional configuration cannot be characterized within the configurational space created  by the basic clusters of the octahedron and tetrahedron.
The point M is on the line joining the
vertices A
2B and A
5B. Hence, at first, a
mixture of the two limiting phases can be attributed to this point. But a homogeneous configuration also lies here, the structure classified Amm2 in the appendix, which has been observed in Monte Carlo simu- lations performed at fixed composition  (Gahn, 1982). This structure is built up with alternating cube planes which are oc- cupied either by pure A or by an equal amount of A and B in an arrangement iden- tical to the cube planes in the L1
1structure,
see Fig. 8-13. It is interesting to notice the structural relationship between this Amm2 structure at point M and the arrangement at K (pure A plus L1
1(AB)): at M the stack-www.iran-mavad.com
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597. ing sequence is one cube plane of pure A
followed by one cube plane of L1
1. If we
increase the stacking distance (i.e., the
number of cube planes of each phase), the
corresponding configurational point moves
from M toward K. At K, the thickness of
each package of cube planes approaches
infinity. The displacement of the configu-
rational point is caused by the decreasing
number of AB pairs that cross the “inter-
faces” between the two structural units.
8.4.1.6 Canonical Energy of lro States
Instead of imposing a given composi-
tion, we may also impose the state of order
(e.g., the most perfect lro) and calculate the
canonical energy for these states as a func-
tion of composition. These are most stable
states under the constraint of homogeneity.
Consider the f.c.c. structure with nearest-
neighbor interactions and the b.c.c. struc-
ture with first and second neighbor interac-
tions as simple examples. In Table 8-1, the
point probabilities for most perfect lro (at
off-stoichiometric composition) are given.
From these probabilities, the numerical
values of the correlation functions for the
different superstructures in their most per-
fect lro state can be derived directly using
Eq. (8-5). These values are given in Table
8-7 for the b.c.c. structures. Introducing
these values into the expression for the ca-
nonical energy
yields the energies of formation of the var-
ious lor phases. In Fig. 8-14a, b, the ener-
gies of formation of b.c.c. structures are
shown for two particular choices of inter-
change energies. It can be seen in Fig. 
8-14a that the canonical energy of the most
stable lro phase for an interchange energy
ratio W
(2)
/W
(1)
= 0.5, namely off-stoichio-
metric D0
3, is a polygon. This means that
all the homogeneous ordered states that can
be formed at every composition are degen-
erate with the two-phase mixtures A + D0
3
and D0
3+ B2. This is different from Fig. 
8-14b where W
(2)
/W
(1)
= –1. The canoni-
cal energy of the most stable lro phase, off-
DDUUNx
N
WW
N
W
N
W
=
= (8-23)
˜
[]
[
]
[
() ( )
()
()
−
−+
+〈〉+〈〉
+〈 〉+〈 〉
+〈〉+〈〉
11
12
1
13 14
23 24
2
12 34
8
86
4
3
8
F
ss ss
ss ss
ss ss
546 8 Atomic Ordering
Figure 8-13.Atomic distribution on four subse-
quent (0 0 1) planes of the superstructures L1
1(AB)
and Amm2 (A
3B). The structure Amm2 is composed
of alternating (0 0 1) planes of pure A and L1
1.
Table 8-7.Numerical values of the correlation func-
tions for the b.c.c. superstructures in their most per-
fect lro state.
B2 D0
3 D0
3 B32
(x
BÙ0.5) (x
BÙ0.25) (0.25Ùx
BÙ0.5) (x
BÙ0.5)
·
s
1Ò 11 1 1
·
s
2Ò 11 11–4 x
B
·s
3Ò1–4x
B 13–8 x
B 1
·
s
4Ò1–4x
B1–8x
B –1 1–4x
B
·s
1s
3Ò1–4x
B 13–8 x
B 1
·
s
1s
4Ò1–4x
B1–8x
B –1 1–4x
B
·s
2s
3Ò1–4x
B 13–8 x
B 1–4x
B
·s
2s
4Ò1–4x
B1–8x
B –1 (1–4x
B)
2
·s
1s
2Ò11 11–4 x
B
·s
3s
4Ò(1–4x
B)
2
1–8x
B –(3–8x
B)1–4x
Bwww.iran-mavad.com 
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598. 8.4 Ground States 547
stoichiometric B2, is a concave curve. In
this instance, the most stable state is a two-
phase mixture of A + B2. From the canoni-
cal energy, we are thus given a hint of
whether the ground state will be two-phase,
or if there is a degeneracy with single-
phase states. The canonical energy for the
f.c.c. lro structures L1
2and L1
0can be de-
rived in a similar way. The result is shown
in Fig. 8-14c for the ratio W
(2)
/W
(1)
=–1.
Here again, the canonical energy of the
most stable lro phase is concave, and only
the two-phase mixtures A + L1
2and
L1
2+L1
0are ground states.
8.4.1.7 Relevant Literature
The ground state analysis of f.c.c. and
b.c.c. alloys with first and second neigh-
bors was done by Kanamori (1966), Rich-
ards and Cahn (1971), Allen and Cahn
(1972, 1973), and de Ridder et al. (1980).
Multiplet interactions were included by
Cahn and Kikuchi (1979), Sanchez and de
Fontaine (1981). The ground states of the
f.c.c. lattice with up to fourth neighbor
interactions were determined by Kanamori
and Kakehashi (1977), fifth neighbor inter-
actions were treated by Kanamori (1979),
Finel and Ducastelle (1984) and Finel
(1984, 1987). The ground states of the hex-
agonal lattice with anisotropic interactions
have been deduced by Crusius and Inden
(1988) and Bichara et al. (1992a) from the
known ground states of the planar hexago-
nal lattice with up to third neighbor interac-
Figure 8-14.Canonical energy D Uof different 
b.c.c. and f.c.c. long-range ordered structures as a
function of composition for fixed values of the inter-
change energies. (a) b.c.c., ordering tendency be-
tween first and second neighbors: W
(1)
=2W
(2)
>0.
(b) b.c.c., ordering tendency between first neighbors
and separation tendency between second neighbors:
W
(1)
=–W
(2)
Û0. (c) f.c.c. ordering tendency be-
tween nearest neighbors: W
(1)
> 0, W
(2)
=0.www.iran-mavad.com
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599. tions (Kudo and Katsura, 1976; Kanamori,
1984).
8.4.2 Effective Cluster Interactions
(ECIs)
Up to this point only pair interactions
have been considered with the exchange
energies W
(k)
as dummy parameters. The
internal energy DU
˜
, with reference to the
pure components in the same crystal struc-
ture, has been written in terms of an expan-
sion of these W
(k)
(Eq. (8-19)). This is a
special case of the more general cluster ex-
pansion (CE) of the internal energy DU
˜
in
terms of the correlation functions:
(8-24)
where nrefers to subclusters of the basic
cluster 
a(n= 1 for point, n= 2 for pair
clusters, etc.) (Sanchez et al., 1984). This
equation is exact if a represents the entire
system. It is expected, however, that Eq.
(8-24) converges rapidly such that the se-
ries can be truncated. For a general discus-
sion of the method see Laks et al. (1992),
de Fontaine (1994) and Sanchez (1996).
The parameters V
nare supposed to em-
brace all possible energy sources that depend
on configuration, including all ground-
state features like electronic energy, long-
range elastic and Coulombic interactions.
But they might also contain configuration-
dependent excitation energies such as those
arising from vibrational and electronic ex-
citations when finite temperatures are con-
sidered, making the model interaction en-
ergies temperature dependent. It should be
appreciated that the parameters V
nrepre-
sent energy contributions, not only from di-
rect interactions within the cluster, but also
from the interactions of much longer range
outside the cluster range. They are thus
called effective cluster interactions (ECIs). 
D
˜
U
N
Vx
nn=
0
a
∑
It may be useful to spell out explicitly
the pure configurational part of the V
n. In
the case of pair interactions the V
nwith
n≥2 are equivalent to the pair exchange 
energies, i.e.,  In this case 
the specific lattice type is only introduced
in terms of the coordination numbers.
In the case of higher-order cluster inter-
actions the lattice type is not only specified
by the coordination numbers of the basic
cluster and its subclusters, but most impor-
tantly by the type of clusters they share
with each other. Subclusters which are not
shared define no new correlations. They are
already taken into account by those of their
parent cluster. We could thus expect that the
V
nof such clusters do not show up in the
expansion, for example triplet correlations
in the tetrahedron expansion of the f.c.c.
lattice. This, however, is not true because
the V
kin the expansion, Eq. (8-24), not only
contain terms from the k-point clusters, but
also terms from higher clusters (n>k).
This becomes transparent if the V
nare
expressed in terms of the cluster exchange
energies, defined as
(for example, W
1234
ABBB
=–4V
ABBB+V
AAAA+
3V
BBBB). It should be noted that, unlike the
pairs, these cluster-exchange energies do
not represent the total energy of the clus-
ters, only successive corrections to the
cluster energies. If total energies were con-
sidered, the sharing of subclusters would
have to be taken into account.
Considering the f.c.c. lattice and a trun-
cation at the regular tetrahedron cluster
(
a= 4), Eq. (8-24) gives:
(8-25)
D
˜
()
()
() () ()
U
N
VVx x
Vx Vx Vxj
jjj
=
AB01
22 33 44+−
+++
WrVV
ij l
r
ij l
ki
l
kk k…
…
……
−+ ∑
12
=
=
V
z
W
n
n
n=
()
()
.
8
548 8 Atomic Orderingwww.iran-mavad.com 
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600. 8.4 Ground States 549
where jindicates specific states, e.g. the
five vertices a–e of the configuration poly-
hedron in Table 8-3. The V
nexpressed in
terms of the cluster exchange energies are
given by 
Taking for example all the triplet terms
W
ijk= 0 does not mean V
3= 0, because
there remain tetrahedron terms.
In the irregular tetrahedron approxima-
tion of the b.c.c. lattice, the cluster expan-
sion of the internal energy is written as:
(8-27)
where the CE parameters V
ncorrespond to: 
V
z
W
z
W
0
1
1
2
2
1
2
88
=
AB
AB AB()
()
()
()
()
DDmm+
−−
D
˜
()
() ()
() () ()U
N
VVx x Vx
Vx Vx Vxj j
jjj
=
AB01 2 2
33 44 55+−+
+++
V
z
W
z
WW
z
WWW
V
z
WW
z
WW
VzW
z
W
0
1
1
1
2
11
1
28
24
64
3
2
1
224
64
22
1
824
=
=
=
AB
triplet
BAA BBA
tetra
BAAA BBAA BBBA
AB
triplet
BAA BBA
tetra
BAAA BBBA
triplet()
()
()( )
()
(
()
()
() ()
DD
DDmm
mm+−
−+
−++
⎛
⎝
⎞
⎠
−− −
−−
+
BAABAA BBA
tetra
BBAA
triplet
BAA BBA
tetra
BAAA BBBA
tetra
BAAA BBAA BBBA
=
(8-26)
=
+
+
−
+−
−+
⎛
⎝
⎞
⎠
W
z
W
V
z
WW
z
WW
V
z
WWW
)
()
()
()
64
3
24
64
22
64
3
2
3
4
These relations show that cluster interac-
tions not only influence the correlations of
their own cluster but also those of the sub-
clusters. The system of equations, Eq. (8-
24), can be solved for the different V
n. The
idea is that the ECIs derived from the ener-
gies of ordered configurations are appli-
cable to any configuration, i.e. for lro, sro
and for the disordered state. This is equiva-
z
WW
WW
z
WW
WW
V
z
WW
1
72
2
2
256
44
24
1
27 2
2
=
triplet
ABA AAB
BBA BAB
tetra
BAAA BABA
BBAA BBBA
AB
triplet
ABA AAB
(
)
(
)
()
(
DD
mm
−+
++
−+
++
−−
⋅+
−−−
−−
++
+
+
⋅−−+
+−
WW
z
WW
V
z
W
z
WW
z
W
V
z
W
z
WWW W
z
WW
V
z
BBA ABB
tetra
BAAA BBBA
AB
triplet
AAB BBA
tetra
BBAA
AB
triplet
ABA AAB BBA BAB
tetra
BABA BBAA
triplet
=
=
=
2
256
88
872
22
256
8
872
22
256
84
2
1
1
3
2
2
4
)
()
()
()
()
()
()
()
()
()
7272
22
256
88
256
44
24
5
()
()
(
)
⋅+−−
+−
−
−+
WWW W
z
WW
V
z
WW
WW
ABA AAB BBA AAB
tetra
BAAA BBBA
tetra
BAAA BABA
BBAA BBBA
=
(8-28
)www.iran-mavad.com
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601. lent to saying, in the pair approximation,
that the W
(k)
can be used for all configura-
tions. Since the only source of error in Eq.
(8-24) is the truncation, it is imperative to
carefully select the truncation.
The ingredients for the determination of
the ECIs are values of DU
˜
(
j), which may
be obtained from various sources (see 
e.g. Zunger, 1994). The simplest approach
is to take experimental data of the corre-
sponding quantities (e.g. Inden (1975a, b,
1977a, b; Oates et al., 1996)). This requires
a truncation at a level imposed by the avail-
ability of data and in most cases does not
provide satisfactory results.
The second approach is to derive DU
˜
(
j)
from first principles calculations, treating
the random alloy by the Coherent Potential
Approximation (CPA) and the electronic
band structure with the Tight-Binding
method (Gautier et al., 1975; Ducastelle
and Gautier, 1976; Treglia and Ducastelle,
1980; de Fontaine, 1984; Sigli and San-
chez, 1988; Sluiter and Turchi, 1989a, b)
or with the Korringa, Kohn and Rostoker
method (Gyorffy and Stocks, 1983; John-
son et al., 1990). The approaches include
the Generalized Perturbation Method and
the Concentration Wave Method. The or-
dering contributions are treated as pertur-
bations of the disordered states. The pertur-
bation method is limited to alloys with sim-
ilar atomic species and provides essentially
pair interactions.
The third approach is the inversion
method of Connolly and Williams (1983)
(Ferreira et al., 1988, 1989; Wei et al.,
1990, 1992; Terakura et al., 1987) . In this
approach special atomic configurations
like in ordered compounds are selected and
their total energy is obtained from direct
electronic structure calculations. 
It has been pointed out by Zunger (1994)
that it is necessary to account for two types
of relaxation, “volume relaxation” when
molar volumes are not constant, and “local
relaxation” when atoms deviate from their
ideal lattice positions. Ferreira et al. (1987,
1988) and Wei et al. (1990) have shown
that, if the volumes depend on composi-
tion but only weakly upon configuration,
the energy of formation can be split into
two additive terms, the elastic energy re-
quired to hydrostatically deform the pure
constituents from their equilibrium vol-
umes to the alloy volume, and the pure
configurational or “chemical” term. The
ECIs can be determined either from a set
of as many ordered compounds as there are
unknowns, or from a larger set, in which
case the ECIs are obtained by a fitting pro-
cedure (Lu et al., 1991, 1995). It must be
emphasized that the ECIs are “effective”
and their values depend on the level of
truncation.
Cluster expansions for systems with
strong lattice relaxations converge slowly
in real space. This has led to an alternative
route, treating the CE in the reciprocal
space (e.g. Laks et al., 1992). 
Finally, it should be mentioned that pair
interactions may also be obtained from the
diffuse scattering of X-rays or neutrons
(for a recent review, see Schönfeld, 1999).
The procedure is the inverse of the CVM
(see Sec. 8.5.1) or the MC technique (see
Sec. 8.5.3). Instead of calculating equilib-
rium configurations using given energy pa-
rameters, experimental data of sro are taken
as input in order to obtain the interaction
parameters. The methods have been pre-
sented by Priem et al. (1989a, b) for CVM,
and Gerold and Kern (1987) and Livet
(1987) for MC.
550 8 Atomic Orderingwww.iran-mavad.com 
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602. 8.5 Phase Equilibria at Finite Temperatures 551
8.5 Phase Equilibria
at Finite Temperatures
Two ways of determining the equilibria
at finite temperatures have proven to be
most useful (see also the chapter by Binder,
2001), CVM introduced by Kikuchi (1950,
1951) and the MC technique (Binder, 1979,
1986; Binder and Stauffer, 1984; Mourit-
sen, 1984; Binder and Heermann, 1988).
The CVM is based on an analytical cal-
culation of the configurational entropy S.
The equilibrium states at constant volume
and temperature are obtained by a mini-
mization of the Helmholtz energy
F(V, T, N
i) = U– TS
for the equilibrium at fixed composition, or
by minimization of the grand potential
W(V, T, m
i) = U– TS– Â
i
m
iN
i
for the equilibrium with exchange of atoms,
i.e., for given values of the chemical poten-
tials.
The MC method simulates the configu-
rations in a computer crystal. At a given
temperature and fixed chemical potentials,
atoms are exchanged with a reservoir of at-
oms with a probability defined in such a
way that the equilibrium state is reached
after a sufficient number of atomic replace-
ments. In the case of fixed composition, at-
oms are selected pairwise and interchanged
with each other according to an equivalent
probability. This method yields the equilib-
rium configuration but not the thermody-
namic functions. On the other hand, we ob-
tain complete and fine-scale information
about the atomic configurations and the
correlations at large distances, the limita-
tion being imposed only by the size of the
computer crystal. Both methods will be ap-
plied in this section.
8.5.1 Cluster Variation Method
In the CVM, the entropy is evaluated ac-
cording to S=k
BlnW, where Wis the
number of arrangements that can be formed
for given values of the correlation func-
tions. The CVM develops an approximate
expression for W, taking into account only
the correlations up to the basic cluster size.
Therefore, the basic clusters are considered
to be uncorrelated independent species.
The first step in determining the entropy is
to write Was the number of possible ar-
rangements of this basic cluster:
(8-29)
where N
ais the total number of a-clusters
in a system with N points, and N
a(s) is the
number of 
a-clusters with a given configu-
ration specified by 
s; the term {a}
N
a
is a
short notation which will be used hereafter
for abbreviation. The ratio N
a(s)/N
awill
be indicated by 
r
a
(s)
.
Eq. (8-29) overestimates the number of
arrangements: two overlapping 
a-clusters
cannot be permuted independently because
they must fit together with their over-
lapping units. This becomes obvious if 
we look at the high temperature limit of Eq.
(8-29), which is W
∞={1}
N
a
, while the 
exact expression of the limit of Wis given
by the number of configurations of N
points: W
∞={1}
N.Therefore, a correction is
needed in order to obtain the correct high-
temperature limit. Correcting for the points
would correspond to the generalized quasi-
chemical approximation (Yang, 1945; Yang
and Li, 1947, 1949). Based on geometrical
considerations, Kikuchi (1950, 1951) intro-
duced a correction in such a way that the
high temperature limit is not only obtained
for the point cluster but also for all subclus-
ters of 
a. A short time later Barker (1953)
W
N
N
N=
!
=
def
a
a
a
a
!
[()]
{}
ss
ss∏www.iran-mavad.com
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603. arrived at the same result by a more mathe-
matical treatment, and the continued inter-
est in the derivation of the entropy formula
is manifested in a number of papers, e.g.,
Hijmanns and de Boer (1955), Burley (1972),
Sanchez and de Fontaine (1978), Gratias et
al. (1982) and Finel (1987). For a dis-
cussion of the hierarchy of the cluster ap-
proximations, see Schlijper (1983, 1984,
1985).
Following Barker, we define m
aas the
number of 
a-clusters per point: N
a=m
aN.
Then we can write (in Stirling’s approxi-
mation):
We now want to correct for the overlap
with the first subcluster 
a–1, which has
one point less than 
a. Due to the indepen-
dent variation of the 
a-clusters, the over-
lapping cluster 
a–1 has also been counted
times, where  is the
number of times the 
a–1 cluster is con-
tained in 
a. The correct number of times
the 
a–1 cluster should be counted, pre-
suming it were an independent species, is
. Therefore, we must correct
Eq. (8-29) as follows:
Using the identity a
a=1, which is valid for
the basic cluster, and a
a–1determined from
the equation
Wcan be written as
W
N
ma
N
ma=({ } ) ({ } )aa
aa aa
−
−−
1
11
ma m mn
aa a aa
a−− −
− −
11 1
1 =
W
N
m N
m
N
mn
N
m
N
mmn=
=
({ } )
({ } )
({ } )
({ } ) ({ } )a
a
a
aa
a
a
aa
a
a aaa
a
−
−
−
−
−
−
−
−
1
1
1
1
1
1
1
N
m({ } )a
a
−
−
1
1
n
a
a
−1
N
mn({ } )a
aa
a
−
−
1
1
{}
()
()
()
(( )
({ } )
() ()
a
rr
a
a
a
a
a
a
aa a
N
m
m
N
m
mN
mN
N
N
=
!
!
=
!
!
=
ss
ss
ss
ss
∏∏
The same reasoning now needs to be fol-
lowed with the next subcluster 
a–2. For
this cluster, the number of arrangements of
the basic cluster 
a, as well as the correction
term for 
a–1, have already been counted.
We can therefore write
with a
a–2determined from the equation
Continuing this reasoning leads to
(8-30)
and to the following recursion scheme for
the exponent a
n, using the identities n
n
n=1
and a
a=1:
(basic) cluster 
a:m
a=m
an
a
aa
a
We thus obtain for the entropy in Stirling’s
approximation
(8-31)
The values of m
aand n
a–m
a
–n
are to be de-
rived by geometrical considerations of the
lattice. This procedure is straightforward
even for large clusters. For the purpose of
illustration, the full set of these values and
the “CVM exponents” a
nare given in Table
8-8 both for the f.c.c. structure in the tetra-
hedron–octahedron approximation and for
the b.c.c. structure in the irregular tetra-
SkNma=
B
=− ∑∑
n
a
nn n n
rr
1 ss
sss s
() ()
ln
a
a
n
aa a
a
a
aa
a
a
aa a
a
a
a a
a
aaa
a
a
n
m
an
an am
n
am−
+
−
++
−− −
−
−
−
−− −
−
−
− −
−
−
−
=
−
−− −
∑
1
2
1
11 1
1
1
1
22 2
2
2
1 1
2
1
1
0:
:
:
mmna
mn a
mmna
mn a mna
mmna
=
=
=  with ÙÙÙ
na
W
N
ma=
=n
a
n
nn
1
∏({ } )
mmnamna
ma
aaa
a
aa a
a
a
aa−
−
− −
−
−
−− +
+
2
2
1 1
2
1
22=
W
N
ma
N
ma
N
ma=({ } ) ({ } )
({ } )aa
a
aa aa
aa
−
×−
−−
−−
1
2
11
22
552 8 Atomic Orderingwww.iran-mavad.com 
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604. 8.5 Phase Equilibria at Finite Temperatures 553
hedron approximation. In the latter case,
the n
a–m
a
–n
have been omitted for the sake of
brevity.
For the entropy, we thus obtain the fol-
lowing:
f.c.c.
=
=
(8-32)
B
BSk
kN
NNN
NN
ij k
ij k ij k
ijkl
ijkl ijkl
ijk
ijk
ln
{}{}{}
{}{}
ln
ln
{}
{}
{}
23 7 1234 12
123 1
2
8
126
81
23 7 23 7
1234 1234
123
…
−
[
+
−
…
…
…
…
…
∑
∑
∑rr
rr
r
lnln
ln ln
{} {}
r
rr rr
123
12 12 11
6
ijk
ij
ij ij
i
ii
+− ]∑∑
The entropy equations (8-32) and (8-33)
are valid for the sro state. In the lro state,
the equivalence of certain clusters is
broken. For example, Eq. (8-33) has to be
written in the lro state with four sublattices
(Fig. 8-2):
b.c.c.
=
=
(8-33)
B
BSk
kN
NNN
NN
ijkl
ijkl ijkl
ijk
ijk ijk
ij
ij ij
ln
{}{}{}
{}{}
ln
ln
ln
{}
{}
{}
1234 12 13
123 1
6
12
3
634
12 1
1234 1234
123 123
12 12
−
[
−
+
+
∑
∑
∑rr
rr
rr
44
13 13 11
{} {}
ln ln
ik
ik ik
i
ii
∑∑ − ]
rr rr
Table 8-8.Numerical values of m
a, n
a–n
a
–m, and the CVM exponents a
n, for the f.c.c. structure in the tetrahe-
dron–octahedron approximation and for the b.c.c. structure in the irregular tetrahedron approximation.
F.C.C. Tetrahedron–Octahedron Fig. 8-4
Cluster 23 … 7 23456 3456 2347 1234 237 123 27 12 1
a =654a4b4c3a3b2a2b1
m
a= 1 6 3122128361
n
6
a=1
n
5
a=6 1
n
a
4a
=3 11
n
a
4b
=12 4 0 1
n
a
4c
=0 0001
n
a
3a
=12 64 2 01
n
a
3b
=8 402401
n
a
2a
=3 2210101
n
a
2b
=12 8 4 5 6 2 301
n
1
a= 6 5 4 4 4 3 3221
a
a=1 00010–101–1
B.C.C. Irregular Tetrahedron Fig. 8-2
Cluster 1234 123 12 13 1
a =432a2b1
m
a=612341
a
a= 1 –1 1 1 –1www.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

605. 8.5.2 Calculation of Phase Diagrams
with the CVM
Once the grand canonical energy U
˜
, Eq.
(8-18), and the entropy, Eq. (8-31) have
been derived, the equilibrium at any tem-
perature and the chemical potentials can be
determined by minimizing the grand poten-
tial 
W=U
˜
–TSwith respect to the configu-
rational variables. Two different tech-
niques of minimization can be found in the
literature: the natural iteration method, 
introduced by Kikuchi (1974), in which
minimization is achieved with respect to
the occupation probabilities of the largest
cluster; and minimization by means of 
the Newton–Raphson and steepest-descent
techniques with respect to the correlation
functions as independent variables (e.g.
Sanchez and de Fontaine, 1978; Mohri et
al., 1985; Finel, 1987). The natural itera-
tion method converges even with starting
values that are far from the equilibrium
values, but it is slow. The second method is
more sensitive to the quality of the starting
values, but it is much faster.
A phase diagram is determined by calcu-
lating the equilibrium state for two phases
and comparing the values of the corre-
sponding grand potentials. In this way, the
most stable phase is obtained. The transi-
tion between different lro states of from lro
to sro can be of first order or higher (usu-
ally second) order. A first-order transition
is detected by the intersection of the grand
potentials of two phases for a certain value
of the chemical potentials (e.g. Kikuchi,
1977b, 1987). This value yields a different
composition for each phase which defines
the tie-line of the two-phase equilibria.
Three-phase equilibria are recognized af-
terwards by the superposition of all the
two-phase equilibria. More sophisticated
techniques have not yet been applied be-
cause phase diagrams of multicomponent
systems beyond ternary have not yet been
analyzed. A second-order transition is
more difficult to define. In this instance,
there is no metastable extension of the ex-
istence range of the most-ordered phase be-
yond the transition point, and the grand po-
tentials meet at this point with a continuous
slope. In such a case, the second Hessian of
the grand potential has to be analyzed
(Kikuchi, 1987). Often this is circum-
vented by showing that there is no meta-
stable extension of the most-ordered phase
within given accuracy limits.
8.5.3 Phase Diagram Calculation
with the Monte Carlo Method
In the MC method the configuration of a
crystal with a limited number of points
(typically several 10
4
to 10
5
) is stored in a
computer. In order to minimize the boun-
dary effects of the finite size of this crystal,
periodic boundary conditions are usually
applied. These boundary conditions must
be consistent with the superstructures we
want to treat so that an integer number of
unit cells fits into the crystal. This deter-
mines both the shape we have to give to the
computer crystal as well as the number of
points. In the b.c.c. structure with first and
second neighbors, the superstructures are
cubic, and there is no difficulty. In the f.c.c.
structure, the superstructures have different
symmetries and the unit cells have differ-
ent extensions in different crystal direc-
tions, as shown in the appendix. The phase
diagram calculation with the MC method is
554 8 Atomic Ordering
Sk
N NN NNN N
NNN N NNNN
=
Bln
{}{}{}{}{}{}{}
{}{}{}{}{}{}{}{}
//
////
1234 12 34 13 14 23 24
123 124 134 234 1 2 3 4
63232111 1
333 314141414
(8-34)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

606. 8.5 Phase Equilibria at Finite Temperatures 555
usually carried out in the grand canonical
scheme, i.e., for a given value of the chem-
ical potentials. This ensures a homogene-
ous single-phase equilibrium state. For this
single-phase state, the configuration can be
analyzed, and a composition determined.
Even in this simple case, problems may 
appear due to defects such as antiphase
boundaries (APBs), which are produced in
the equilibration process in the same man-
ner as in real alloys (see, e.g., Gahn, 1986;
Ackermann et al., 1986; Crusius and Inden,
1988). 
The Monte Carlo process can be started
from any configuration and composition,
even from the pure components. An arbi-
trary site nis selected at random and the
atom ion this position is replaced by an
atom jaccording to a transition probability
w
n
iÆjwhich has to fulfill certain criteria in
order to guarantee the convergence toward
the equilibrium state from any initial con-
figuration (Chesnut and Salsburg, 1963;
Fosdick, 1963; Binder, 1976). A sufficient
condition for this convergence is the fulfill-
ment of the detailed balance equation and
of normalization of the transition probabil-
ity:
(8-35a)
(8-35b)
Different expressions for the transition
probability which fulfill this criterion have
been suggested. The choice is made ac-
cording to the most efficient expression 
in terms of computer time. The following
expression fulfills the conditions of Eq. 
(8-35):
(8-36)
w
kT
j
n
i
n
kT
j
n
i
n
n
ij
j
K
→
−
⎧
⎨
⎩
⎫
⎬
⎭
−
⎧
⎨
⎩
⎫
⎬
⎭
⎛
⎝
⎜
⎞
⎠ ⎟
⎡
⎣
⎢
⎤
⎦ ⎥
−
⎧
⎨
⎩
⎫
⎬
⎭
−
⎧
⎨
⎩
⎫
⎬
⎭
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
∑
=
B
= B
exp
exp
1
1
1
WW
WW
rr
n
i
n
ij
n
j
n
ji
j
K
n
ijww
w
→→
→
∑
=
=
=1
1
where  is the grand canonical energy 
when an atom of kind ioccupies site n. Be-
cause the sum of the probabilities over all
components adds up to 1, the interval [0, 1]
can be subdivided into corresponding inter-
vals. The atom ion site nis then replaced
by the atom j , which corresponds to the
interval selected by a random number be-
tween 0 and 1. If the random number se-
lects the interval corresponding to i, no ex-
change is made. In a Monte Carlo simula-
tion, the above-mentioned steps have to be
performed many times, and it is necessary
to use efficient programming techniques
for data storage and saving of computer
time (for general aspects see Binder (1976,
1979, 1985), Binder and Stauffer (1984);
for multispin coding see Jacobs and Rebbi
(1981), Zorn et al. (1981) and Kalle and
Winkelmann (1982)).
8.5.4 Examples of Prototype Diagrams
The following examples represent proto-
type diagrams. The interchange energies
W
(k)
used in the calculations are selected in
such a way that the phase diagrams display
typical features. Calculations for real al-
loys will be presented in the next section.
We will start with the systems treated in the
previous sections.
8.5.4.1 F.C.C. Structure,
First Neighbor Interactions
In this instance we take the case of an 
ordering tendency in the first shell, i.e.,
W
(1)
> 0, W
(2)
= 0. The phase diagram, as
obtained by CVM in the tetrahedron–octa-
hedron approximation (Finel and Ducas-
telle, 1986), together with the results from
the MC method (Ackermann et al., 1986;
Diep et al., 1986; Gahn, 1986), are shown
in Fig. 8-15. The two diagrams do not coin-
W
i
n
⎧
⎨
⎩
⎫
⎬ ⎭www.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

607. cide quantitatively, but their main features
agree qualitatively. In particular, they agree
as to the existence of a triple point at finite
temperature, i.e. at 
t
tri
CVM= k
BT
tri
CVM/(W
(1)
/4) ≈1.5
(Finel and Ducastelle, 1986) and at
t
tri
MC≈0.8 (Ackermann et al., 1986) and 0.9
(Diep et al., 1986). This point gave rise to
some controversy (discussed by Kikuchi,
1986), caused by an earlier MC study 
(Binder, 1980; Binder et al., 1981) which
indicated that the phase boundaries extra-
polate to 0 K, and that a triple point does
not exist. The existence of this triple point
is now well confirmed and has been further
corroborated by the studies of Lebowitz et
al. (1985) and Finel (1994). The diagram in
Fig. 8-15 replaces that of the earlier MC by
Binder. The MCs were not performed at
sufficiently low temperatures to detect the
P4/mmm phase obtained in the CVM 
(called L¢ by Finel, 1984).
The CVM diagram in the tetrahedron–
octahedron approximation, Fig. 8-15, dif-
fers only slightly from that previously cal-
culated in the tetrahedron approximation
(van Baal, 1973; Kikuchi, 1974), which
gave a triple point at 
t
tri
CVM≈1.6. The
higher-order cluster approximation does
not lead to a strong shift of the triple point
to lower temperatures, contrary to an ear-
lier result by Sanchez et al. (1982) who
found 
t
tri
CVM≈1.2. The difference results
only from certain approximations in the
numerical treatment made by Sanchez et
al., which were avoided by Finel (1987). In
Finel’s work, a more sophisticated CVM
calculation was also made using the tetra-
hedron–octahedron for the ordered phases
and the quadrupole tetrahedron for the sro
states. In that approximation. Finel ob-
tained complete agreement with the MC
work.
556 8 Atomic Ordering
Figure 8-15.F.c.c. structure: calculated prototype phase diagrams for the case of nearest-neighbor interactions,
W
(1)
> 0, W
(2)
= 0. (a) CVM calculation in the tetrahedron–octahedron approximation (Finel and Ducastelle,
1986), (b) Monte Carlo simulation (Ackermann et al., 1986).www.iran-mavad.com 
+ s e  l ⎨'4 , kp e  r  i ⎨&s ! 9 j+  N   0 e

608. 8.5 Phase Equilibria at Finite Temperatures 557
8.5.4.2 F.C.C. Structure, First and
Second Neighbor Interactions
In this section, the case of an ordering
tendency in both shells will be treated first,
namely, W
(1)
> 0 and W
(2)
/W
(1)
= 0.25. The
results of the tetrahedron–octahedron
CVM (Sanchez and de Fontaine, 1980) and
of MC simulations (Bond and Ross, 1982)
are shown in Fig. 8-16. The two diagrams
agree fairly closely. Phase diagrams for
other values of the ratio W
(2)
/W
(1)
are
given by Mohri et al. (1985) and Binder 
et al. (1983).
Fig. 8-17 shows the results for an order-
ing tendency between first neighbors and a
separation tendency between second neigh-
bors, W
(1)
> 0, W
(2)
=–W
(1)
. This situation
corresponds to the calculation presented
previously, in Fig. 8-14c. As expected
Figure 8-16.F.c.c. struc-
ture: calculated prototype
phase diagrams for the case
of first and second neighbor
ordering interactions,
W
(1)
=4W
(2)
> 0. (a) CVM
calculation in the tetrahe-
dron–octahedron approxi-
mation (Sanchez and de
Fontaine, 1980). (b) Monte
Carlo simulation (Bond and
Ross, 1982).
Figure 8-17.F.c.c. structure:
calculated prototype phase dia- grams for the case of an order- ing tendency between first neighbors and a separation ten- dency between second neigh- bors, W
(1)
=–W
(2)
> 0. (a) CVM
calculation in the tetrahedron– octahedron approximation  (Mohri et al., 1985). (b) Monte Carlo simulation (Binder et al., 1983).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

609. from this diagram the low temperature
states are two-phase states and the phase
boundaries extrapolate to the pure compo-
nents and to the stoichiometric ordered
phases. A complete series of prototype 
diagrams calculated with the tetrahedron–
octahedron CVM is presented by Mohri et
al. (1985).
The case of a separation tendency be-
tween first and second neighbors has been
analyzed with the MC technique: Kutner 
et al. (1982) analyzed the case W
(1)
<0,
W
(2)
= 0 with the grand canonical simula-
tion, while Gahn et al. (1984) analyzed the
cases W
(2)
= 0 and W
(2)
≠0 with a spe-
cial canonical simulation. The results for
W
(2)
= 0 are identical in both treatments.
The resulting miscibility gaps are shown in
Fig. 8-18 (on a reduced scale in order to
show the variation in shape). It is found
that the effect of second neighbor interac-
tions on the shape of the miscibility gap is
small. The shape obtained differs signifi-
cantly from the miscibility gap which is
usually calculated with the regular solution
(i.e., point approx.) model. 
8.5.4.3 B.C.C. Structure, First and
Second Neighbor Interactions
A series of prototype phase diagrams,
calculated with the tetrahedron CVM for
varying strengths of ordering tendency in
both neighbor shells, was first presented by
Golosov and Tolstick (1974, 1975, 1976).
Simultaneously, Kikuchi and van Baal
(1974) presented a diagram corresponding
to the ratio W
(2)
/W
(1)
= 0.5, which is close
to the situation encountered in Fe–Si and
Fe–Al. Figs. 8-19a, b and c display a series
of diagrams calculated with the tetrahedron
CVM and with the MC method (Acker-
mann et al., 1989) for different strengths
and signs of the interchange energies. The
results of both methods are in good agree-
ment. In these diagrams, second-order tran-
sitions are indicated by a hachure. These
second-order transitions turn into first-or-
der transitions at tricritical points, and the
topology of the phase boundaries close to
these points exhibit the characteristics de-
rived by Allen and Cahn (1982).
8.5.4.4 Hexagonal Lattice, Anisotropic
Nearest-Neighbor Interactions
The ordering reactions in the hexagonal
crystal structure have been studied with 
the MC method (Crusius and Inden, 1988;
Bichara et al., 1992b). In order to simulate
the situation for c/a≠1.633, i.e., for non-
close packing, the interchange energies
between nearest neighbors within a basal
plane, W
(11)
, and between two such planes,
W
(12)
, were given different values. The 
resulting phase diagrams for two sets of
interchange energies are shown in Figs. 
558 8 Atomic Ordering
Figure 8-18.Miscibility gaps according to MC cal-
culations for various values of the interchange ener-
gies between first and second neighbors. The dia-
gram is symmetric with respect to the equiatomic
composition, W
(1)
<0, W
(2)
= 0 (Kutner et al., 1982).
W
(2)
= 0 and W
(2)
≠0 (Gahn et al., 1984).www.iran-mavad.com 
+ s e  l ⎨'4 , kp e  r  i ⎨&s ! 9 j+  N   0 e

610. 8.5 Phase Equilibria at Finite Temperatures 559
8-20a and b. The phase A
2B is a two-
dimensional structure built with three sub-
sequently ordered planes, but with no cor-
relation between them. Therefore a two-di-
mensional characterization has been given
in Sec. 8.7 in addition to the three-dimen-
sional one that is to be considered for cases
with W
(12)
≠0 (see Appendix, Table 8-13).
In the case W
(12)
= 0, the phase A
2B is
the only stable superstructure. The transi-
tion A
2B´A3 is second order in this in-
stance. For 
m*=0 (x
B= 0.5), no phase tran-
sition has been observed down to the re-
duced temperature 
t=k
BT/(W
(11)
/4) = 0.6.
The extrapolation of the transition temper-
ature goes to 0 K at this composition. This
is consistent with the exact solution which
Figure 8-19.B.c.c. structure: calculated prototype phase diagrams for ordering or separation tendencies
between first or second neighbors. The lines correspond to the CVM calculation in the tetrahedron approxima-
tion, the points correspond to the Monte Carlo simulation (Ackermann et al., 1989). The hachure indicates a 
second-order transition. (a) W
(1)
=2W
(2)
> 0, (b) W
(1)
=–W
(2)
> 0, (c) W
(1)
=–W
(2)
<0.www.iran-mavad.com
+ s e l ⎨'4 , kp e r i ⎨&s ! 9 j+ N 0 e

611. is known for m* = 0 (Houtappel, 1950; Ne-
well, 1950; Wannier, 1950). The diagram
for W
(12)
=W
(11)
corresponds to the hexag-
onal close-packed structure, see Bichara 
et al. (1992b). This diagram is not shown
because it is exactly the same as the one in
Fig. 8-15, except that the phases L1
2and
L1
0have to be replaced by the phases D0
19
and B19, respectively. The reason for this
is that the f.c.c. and hexagonal close pack-
ings cannot be distinguished by nearest-
neighbor interactions only.
8.5.5 The Cluster Site Approximation
(CSA)
Notwithstanding the successes of the
CVM in the calculation of phase diagrams
and thermodynamic properties, a major
disadvantage is the large number of con-
figurational variables, and thus of non-
linear equations, to be solved in order to
minimize the Helmholtz energy. The num-
ber of independent variables in a K-com-
ponent system and an r-site CVM is 
K
r
– 1. This high number increases strongly
with the size of the basic cluster, even for
binary systems. Real systems, however,
may involve many more than three compo-
nents.
The CSA suggested by Oates and Wenzl
(1996) is a revival of the quasi-chemical
tetrahedron approximation (Yang and Li,
1947). A system with Npoints is decom-
posed into N
aclusters of type ain such a
way that the clusters share only points. The
important result of this assumption is that
only point correlation functions are used in
the Helmholtz energy minimization. The 
a
cluster probabilities are obtained after the
calculation from the quasi-chemical equi-
librium between the atoms (points) and
molecules (clusters). If 
acontains rpoints,
the number of independent variables is
(K–1) (r–1). Thus with increasing cluster
size or number of components the number
of independent variables in the CSA is sig-
nificantly smaller than in the case of CVM.
This is the main advantage.
Following the concept of the CVM, the
entropy of the CSA in the r-site cluster ap-
proximation can be written
(8-37)
Sk
r
SrS
N
r
N
r
N
=
=
B
=
=ln
({ } ) { }
({ } )
()
12
1
1
1
1
…
−− ∏
∏
g
n
n
g
g
n
n
gg
560 8 Atomic Ordering
Figure 8-20.Hexagonal structure: calculated proto-
type phase diagrams for the case of anisotropic near-
est-neighbor interactions, W
(11)
within the basal
plane and W
(12)
between basal planes. Monte Carlo
calculations (Crusius and Inden, 1988). The hachure
indicates a second-order transition. (a) W
(12)
>0,
W
(12)
/W
(11)
= 0.8, (b) W
(11)
> 0, W
(12)
=0.www.iran-mavad.com 
+ s e  l ⎨'4 , kp e  r  i ⎨&s ! 9 j+  N   0 e

612. 8.6 Application to Real Systems 561
where gis the coordination number of tet-
rahedra per lattice point. In the case of the
f.c.c. lattice and tetrahedron cluster this
number is 
g=1, i.e. half the value used in
the CVM.
The principal reason for the lack of at-
tention to the CSA is its inability to obtain
the correct topology of prototype phase di-
agrams as obtained from CVM or MC cal-
culations. The tetrahedron approximation
of the f.c.c. lattice is a prominent example.
The CSA with 
g=1 gives lro regions L1
2
and L1
0around the stoichiometric compo-
sitions, but between these regions the dis-
ordered phase is predicted to be stable
down to 0 K. Oates et al. (1999) have
shown that a simple correction of the value
of 
gallows us to compensate for this. The
authors were able to reproduce the f.c.c.
prototype phase diagram of the tetrahedron
CVM by taking 
g=1.42.
Several successful results have been ob-
tained with the CSA; those for the Cu–Au
system will be mentioned here. Using the
value 
g=1.42 as a constant entropy correc-
tion term for the tetrahedron treatment of
f.c.c. alloys, Oates et al. (1999) applied the
CSA to the Cu–Au system using the tetra-
hedron interactions from Kikuchi et al.
(1980). They obtained exactly the same
phase diagram as Kikuchi with the CVM.
The diagram is shown in Fig. 8-21a. Fur-
thermore, taking the many-body interac-
tions and also the elastic energy arising
from atomic size mismatch from Ferreira et
al. (1987), Oates et al. obtained again the
same phase diagram as Ferreira with the
CVM, see Fig. 8-21b. This diagram also
shows the effect of the elastic terms: they
contribute positive corrections to the con-
figurational energy, thus removing the de-
generacy of the ground states. Therefore, at
0 K, the phase boundaries meet at the stoi-
chiometric compositions and go to the pure
components, contrary to the prototype dia-
grams in Figs. 8-15, 8-16 and 8-19a where
such effects were not included.
These results indicate that the CSA may
be a very useful tool when it comes to tech-
nological problems. It remains to be seen
whether the CSA can be applied as suc-
cessfully to other structures and to larger
clusters.
8.6 Application to Real Systems
In contrast to prototype systems, real
systems exhibit very complex properties
Figure 8-21.Cu–Au phase diagrams calculated
with the CSA using the constant entropy correction
factor 
g= 1.42 (Oates et al., 1999). a) Calculated
without any size mismatch using the pair and CuCu-
CuAu and CuAuAuAu tetrahedron interactions de-
rived by Kikuchi et al. (1980) using the tetrahedron
CVM. The phase diagram is the same as that of the
CVM. b) Calculated with size mismatch using the
many body interactions and the elastic energy terms
derived by Ferreira et al. (1987) using the tetrahe-
dron CVM. The phase diagram is the same as that of
the CVM.www.iran-mavad.com
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613. even at T= 0 K. With the progress made in
first-principles calculations, it has become
obvious that higher-order pair and multisite
cluster interactions have to be used in the
configurational part of enthalpy, and that it
is mandatory to take lattice and local relax-
ations into account. This will be illustrated
using the Au–Ni system as an example.
When it comes to T> 0 K the cluster inter-
actions have to embrace all configuration-
dependent excitation energies such as arise
from thermal vibrations and electronic ex-
citations. The inclusion of these effects
means that the cluster interactions become
temperature dependent. There is still some
disagreement as to whether vibrational
contributions can (Ozolin¸sˇ et al., 1998b;
van de Walle et al., 1998) or cannot (Craie-
vich and Sanchez, 1997; Craievich et al.,
1997a, b) be included in the cluster expan-
sion.
However, even the apparently most care-
ful first-principles calculations (Ozolin¸sˇ et
al., 1998a) remain insufficiently accurate
for technological purposes. For the fore-
seeable future it seems clear that simpler
approaches will continue to be important,
e.g. for technological phase diagram calcu-
lations and also for other applications re-
lated to the thermodynamic properties. The
systems Fe–Al and Ni–Al will be taken
here as examples for calculations of phase
diagrams and of the thermodynamic factor
for diffusion. 
Surveys of the abundant literature de-
scribing applications to mainly binary al-
loy systems can be found in the reviews by
Inden and Pitsch (1991) and by de Fontaine
(1994).
At this stage multicomponent systems
present apparently insurmountable prob-
lems for first-principles calculations. This
even holds for ternary systems. Rubin and
Finel (1993) studied ternary Ti–Al–X
(X=W, Nb, Mo) systems with first and sec-
ond neighbor pair interactions and used a
power series expansion in composition for
the disordered state. No ternary interac-
tions were included. The agreement of the
results for the limitrophe binary systems
with experimental data is satisfying only in
a few of the cases studied, and very little is
presented with respect to ternary isother-
mal sections. McCormack et al. (1996,
1997) studied the systems Cd–Ag–Au
(1996) and Cu–Al–Mn (1997) using first
and second neighbor pair interactions, but
failed to present phase equilibria for ter-
nary and limitrophe binary systems that
could be checked by experiments. In 
the following the two ternary systems
Fe–Ti–Al and Fe–Co–Al will be dis-
cussed. The Fe–Co–Al system will be
treated as a magnetic system with spin 1/2
given to Fe, Co and Al. This leads in fact to
a six-component system. 
In multicomponent systems the number
of points within a chosen basic cluster may
not be large enough to accommodate all
components. Then the question arises as to
what extent this is detrimental to the qual-
ity of the approximation. In order to give
some idea of this effect, a magnetic spin
7/2 system with f.c.c. structure has been
analyzed by Schön and Inden (2001) using
the tetrahedron CVM and Monte Carlo
simulations. This spin problem is equiva-
lent to an eight-component alloy problem.
The calculation of phase equilibria and
thermodynamic properties of multicompo-
nent systems is of paramount importance
for technological design and materials de-
velopment. The use of both a cluster ex-
pansion for the energy and the CVM for
configurational entropy introduces severe
computational problems because of the
large number of cluster types that must be
considered in both formalisms. It also has
to be emphasized that in practical applica-
tions with coexisting phases of different
562 8 Atomic Orderingwww.iran-mavad.com 
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614. 8.6 Application to Real Systems 563
crystal structures, the calculation of the
atomic configurations has to be performed
for each of the structures, and within each
structure with the same level of approxima-
tion for all subsystems in order to achieve
consistency. That is to say, isolated sophis-
ticated treatments of particular systems
will provide deep insight into the physics,
but they are only of limited use for solving
technological problems.
Recently, the quasi-chemical tetrahedron
approximation has been adapted to over-
come the problems of obtaining the cor-
rect topology of order–disorder phase dia-
grams for f.c.c. systems and to treat lattice
relaxations and excitations. This method
seems to offer the possibility of handling
order–disorder effects with sufficient qual-
ity within a format that can be used in ther-
modynamic databases of multicomponent
systems. The Cd–Mg system will be taken
as an example for illustration, because it
has been extensively analyzed: all possible
contributions to the ground states have
been treated by first-principles calculation,
and the phase diagram has been calculated
by CVM (Asta et al., 1993).
8.6.1 The Au–Ni System
This system has been analyzed inten-
sively over the last decade (Renaud et al.,
1987; Eymery et al., 1993; Wolverton and
Zunger, 1997; Wolverton et al., 1998; 
Ozolin¸sˇ et al., 1998; Colinet and Pasturel,
2000). It has attracted much interest because
of a phase separation tendency and positive
enthalpies of mixing at low temperatures,
ordering type of short-range order at high
temperatures, and a large lattice mismatch
of about 15% between the constituents. It
may be considered a key system for check-
ing the quality of the theoretical methods.
EXAFS experiments (Renaud et al.,
1987) have shown that the atoms are
strongly displaced relative to the regular
undistorted lattice. The distribution of
Au–Au distances was found to be narrow
and weakly asymmetric, while those for
Ni–Ni were found to be widely distributed
and highly asymmetric. Eymery et al.
(1993) came to the same conclusion on the
basis of their theoretical work. From the
calculated average nearest-neighbor dis-
tances, partial tetrahedral volumes were
calculated as a function of composition.
The results are shown in Fig. 8-22. The size
Figure 8-22.Variation of the
average tetrahedral volumes
Au
4, Au
3Ni, AuNi, AuNi
3and
Ni
4in random Au
1–xNi
xsolid
solutions at T = 0 K versus Ni
concentration [redrawn from
Eymery et al. (1993)]. The vol-
umes were calculated from the
calculated average nearest-
neighbor distances. The differ-
ence between the various vol-
umes and their variation with
composition reflects the lattice
relaxations. The calculations
were based on a tight-binding
second moment approximation.www.iran-mavad.com
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615. distribution of the tetrahedral volumes was
also calculated. The results for the equi-
atomic composition are shown in Fig. 8-23.
The local relaxations become evident by
the deviation of the average positions of
these distributions from the corresponding
average volumes in Fig. 8-22, which are
shown in Fig. 8-23 by the arrows. These re-
sults clearly indicate that both lattice and
local relaxations must not be ignored. This
fact becomes also visible in the enthalpy of
formation of the random alloys differing by
a factor of two between unrelaxed and re-
laxed states.
Wolverton et al. (1997, 1998) considered
the Au–Ni system and analyzed the ques-
tion of up to what order the pair interaction
scheme has to be driven in order to repro-
duce experimental findings. The CE was
separated into three parts: the pair interac-
tions with arbitrary distance summed in the
reciprocal space representation (Laks et al.,
1992), the multi-atom interactions in real-
space representation, and the constituent
strain energy. The authors concluded that
pair interactions up to about the 15th shell
are needed. Ozolin¸sˇ et al. (1998a, b) exam-
ined the same question for the series of no-
ble metal alloys Cu–Au, Ag–Au, Cu–Ag
and Au–Ni. While in Ag–Au and Ag–Cu
the first three neighbor pair interactions are
dominant, the same is not true for the other
systems, particularly not for Au–Ni, where
not only pair interactions up to the 10th
shell, but also triplet and four-point cluster
interactions in increasing distances have to
be taken into account.
Most recently, Colinet et al. (2000) took
up to fourth neighbor pair and triplet inter-
actions into account, the tetrahedron inter-
actions turning out to be almost negligible
in their treatment. They achieved good
agreement in the entropy of mixing, fairly
good agreement in the enthalpy and Gibbs
energy of mixing and calculated the phase
diagram by means of the tetrahedron–octa-
hedron CVM. The comparison between the
calculated and experimental miscibility
gap is shown in Fig. 8-24.
564 8 Atomic Ordering
Figure 8-23.Histogram of  the calculated partial tet-
rahedral volumes Au
4, Au
3Ni, AuNi, AuNi
3and Ni
4
in a random alloy at T= 0 K with x
Ni= 0.5 [redrawn
from Eymery et al. (1993)]. The volume distributions
are broad and partially asymmetric. The average vol-
umes, taken from Fig. 8-22, are shown by the arrows.
The difference between these average values and the
centers of gravity of the distributions is remarkable
showing the importance of local relaxations.
Figure 8-24.Miscibility gap of the Au–Ni system.
Experiments from Bienzle et al. (1995). The calcula- tions were performed using pair and triplet interac- tions in the tetrahedron–octahedron CVM (Colinet et al., 2000).www.iran-mavad.com 
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616. 8.6 Application to Real Systems 565
8.6.2 The Thermodynamic Factor
of Ordered Phases
8.6.2.1 The B.C.C. Fe–Al System
Iron aluminides such as Fe
3Al and FeAl
have received considerable attention as
candidates for high temperature structural
materials due to their low cost, high
strength and good oxidation resistance.
Schön and Inden (1998) assessed the sys-
tem in order to derive ECIs from experi-
mental enthalpies of formation, corrected
for b.c.c reference states and converted into
the paramagnetic state: 
j DH
j
[J/mol of atoms]
Fe
3Al (D0
3)–18650
FeAl (B2) –27 940
FeAl(B32) –21 570
FeAl
3(D0
3)–1 8650
Introducing these enthalpies and the values of the correlation functions of the configu- rations (see Table 8-5) into Eq. (8-24), the
following parameters are obtained (in units of k
BK)
V
0= –2190,V
1= 0,V
2= 1680,
V
3= 457,V
4= 0,V
5= 52.5
or equivalently
(8-38)
W
(1)
= 1680,W
(2)
=740,W
1234
FeAlFeAl
=–140
With these parameters the irregular tetrahe- dron CVM yields the b.c.c. phase diagram shown in Fig. 8-25. The calculated critical temperatures match the high temperature experimental data. However, at lower tem- peratures, the tricritical point and the two- phase region A2 + B2 are not obtained and the agreement with the experimental phase boundaries A2 + D0
3is poor. This is not
surprising in view of the large relaxation effects, which are to be expected in this system as seen from the variation of lattice parameter as a function of composition and of state of order, Fig. 8-26. On the other hand, recent experimental data on chemical potential measurements at 1000 K (Kley- kamp and Glasbrenner, 1997) are very well
Figure 8-25.The b.c.c. phase dia-
gram of Fe–Al calculated with the
irregular tetrahedron CVM using
the parameters given in Eq. (8-38)
(Schön and Inden, 1998). Second-
order transitions are indicated by
broken lines. Three temperatures
are indicated at which the thermo-
dynamic factor of diffusion is
shown below in Fig. 8-28. Experi-
mental points: neutron diffraction
and c
pfrom Inden and Pepperhoff
(1990), dilatometry from Köster
and Gödecke (1980), TEM and
DTA from Ohnuma et al. (1998a).www.iran-mavad.com
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617. reproduced by the calculations, see Fig. 
8-27.
A very important quantity for the treat-
ment of diffusion is the thermodynamic
factor defined for a binary system A–B as
This quantity is smooth in a random solid
solution, but shows a complicated variation
F=
d
d
BB
Bx
xm
in ordered alloys. Fig. 8-28 shows this vari- ation at three different temperatures. Close to stoichiometric compositions of ordered phases, 
Fvaries dramatically. The amount
of variation depends on the degree of lro, reaching up to an order of magnitude at least in almost fully lro states, as observed at T= 650 K for B2 at the composition
x
Al= 0.5. At second-order transition points
Fchanges discontinuously.
566 8 Atomic Ordering
Figure 8-26.Variation of the
lattice parameter of Fe–Al alloys
as a function of composition and
state of order. Experiments: Lihl
and Ebel (1961).
Figure 8-27.Chemical potential
of Fe and Al at 1000 K in b.c.c. Fe–Al alloys calculated with the tetrahedron CVM using the  parameters given in Eq. (8-38)  (Schön and Inden, 1998). Experi- mental data from Kleykamp and Glasbrenner (1997).www.iran-mavad.com 
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618. 8.6 Application to Real Systems 567
8.6.2.2 The F.C.C. Ni–Al System
The variation of the thermodynamic fac-
tor is even more pronounced in the Ni–Al
system because there the order–disorder
transitions occur at much higher tempera-
tures than in Fe–Al, as shown in the cal-
culated f.c.c. phase diagram in Fig. 8-29 
(Schön and Inden, 1998). The ECIs were
derived from the ground-state energies of
L1
2–Ni
3Al, L1
2–NiAl
3, and L1
0–NiAl,
obtained by Pasturel et al. (1992) from
first-principles calculations:
j DH
j
[J/mol of atoms]
Ni
3Al (L1
2) –48 000
NiAl (L1
0)– 56000
NiAl
3(L1
2) –22 000
Introducing these enthalpies and the values of the correlation functions of the configu- rations (see Table 8-3) into Eq. (8-24), the following parameters are obtained (in units of k
BK)
V
0= –4630,V
1= –1563,V
2= 5051,
V
3= 1563,V
4= –421
or equivalently
W
(1)
= 3370,W
1234
NiAlAlAl
= – 4810,
W
1234
NiNiNiAl
= 1440 (8-39)
The calculated thermodynamic factor of
diffusion is shown in Fig. 8-30 for the two
temperatures indicated in Fig. 8-29. Due to
the high degree of lro, the variation of 
Fin
the range of stoichiometric composition is
Figure 8-28.Thermodynamic factor of diffusion in b.c.c. Fe–Al alloys, calculated as a function of composi-
tion at three temperatures, 650 K, 1000 K and 1400 K. The calculations were performed with the irregular 
tetrahedron CVM using the parameters given in Eq. (8-38) (Schön and Inden, 1998). At second-order transi-
tions, 
Fchanges by a step. Close to the stoichiometric compositions x
Al= 0.25 and 0.5 the value of Fincreases
as a function of the degree of lro. For example, the temperature of T= 650K is only slightly below the D0
3/B2
transition temperature, see Fig. 8-25. Therefore, at x
Al= 0.25 the increase is small compared to x
Al= 0.5 where
Fincreases by more than an order of magnitude because the distance from the B2/A2 transition temperature is
large and the degree of lro almost maximum. With increasing temperature this effect becomes smaller.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

619. particularly pronounced. Variations of the
same order of magnitude are obtained for
the B2 phase.
It is worth mentioning that these dra-
matic variations have repercussions in the
composition profiles of diffusion couples.
An example is shown in Fig. 8-31, where a
contrast has been observed in the micro-
graph within the B2 phase region as it is
usually observed at phase boundaries. The
electron microprobe analysis reveals a
steep change of composition at this boun-
dary, which again could be misinterpreted
as a tie-line. This is an important fact to be
realized, because phase diagram determi-
nations in multicomponent systems are
most conveniently performed by means of
diffusion couple experiments.
568 8 Atomic Ordering
Figure 8-29.Phase diagram
of the f.c.c. Ni–Al system
calculated with the regular
tetrahedron CVM (Schön
and Inden, 1998) using the
parameters given in Eq. 
(8-39). Two temperatures are
indicated for which the ther-
modynamic factor has been
calculated, Fig. 8-30.
Figure 8-30.Thermody-
namic factor of diffusion in f.c.c. Ni–Al alloys, calcu- lated as a function of com- position at two tempera- tures, 1500 K and 2350 K (Schön and Inden, 1998). The calculations were per- formed with the regular tetrahedron CVM using  the parameters given in Eq. (8-39). At the stoichiomet- ric compositions, 
Fin-
creases by more than an order of magnitude.www.iran-mavad.com 
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620. 8.6 Application to Real Systems 569
8.6.3 Ternary Systems
The b.c.c. states of the two systems
Fe–Ti–Al and Fe–Co–Al will now be
treated by taking the energy description
from the limitrophe binaries in order to see
what can be obtained by extrapolating from
the binaries into the ternary system without
ternary interaction terms. 
8.6.3.1 B.C.C. Fe–Ti–Al
Ohnuma et al. (1998b) studied ordering
and phase separation in the b.c.c. phase of
the ternary system, with particular empha-
sis on ternary miscibility gaps between dis-
ordered and ordered phases. The theoreti-
cal analysis was based on the irregular tet-
rahedron CVM. The analysis of the Fe–Ti
and Ti–Al binary systems led to the energy
parameters given in Table 8-9. The param-
eters for Fe–Al were taken as given in Eq.
(8-38).
The calculated isothermal sections at
1173 and 1073 K are shown in Figs. 8-32
and 8-33. From the metallurgical point of
Figure 8-31.Diffusion couple Ni
77Al
23/Ni
40Al
60
annealed at 1273 K for 100 h (Kainuma et al., 1997).
a) Microstructure showing a change in contrast
within the B2 phase field. The regions labelled b
1
and b
2do not represent two different phases. The
pseudo-boundary between b
1and b
2comes from a
steep change in composition. b) Composition profile
measured by electron probe microanalysis.
Table 8-9.Atomic exchange energy parameters in
units of k
BK (Ohnuma et al., 1998b).
A–B W
AB
(1) W
AB
(2) W
1234
ABAB
Ti–Fe 1580 –1050 1800
Al–Ti 2420 1200 0
Figure 8-32.Calculated isothermal section of the
b.c.c. Fe–Ti–Al system at 1173 K using the energy parameters in Table 8-9. Second-order transitions are shown as broken lines. The enlarged part of the sec- tion shows the good agreement obtained between ex- periments and calculation (Ohnuma et al., 1998b).www.iran-mavad.com
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621. view it is interesting to see the opening of
ternary miscibility gaps between ordered
and disordered phases. This allows us to
produce coherent two-phase equilibria with
interesting mechanical properties. Guided
by the calculations, Ohnuma et al. (1998b)
also performed experiments in order to
confirm the existence of the two-phase
fields experimentally. The results are
shown in the enlarged parts of the sections
of Figs. 8-32 and 8-33. The calculated tie-
lines agree very well with the experimental
results.
8.6.3.2 B.C.C. Ferromagnetic Fe–Co–Al
(six-component system with spin 1/2 
for Fe, Co and Al)
This system has been studied by Colinet
et al. (1993) in the irregular tetrahedron 
approximation. The binary system Fe–Al
was discussed in Sec. 8.6.2.1, but without
taking magnetic effects into account. Be-
cause the Curie temperature of metastable
b.c.c. Fe–Co alloys goes up to about
1500 K, the magnetic effects cannot be 
disregarded in this ternary system. A spin
1/2 treatment has been taken in this in-
stance. The magnetic exchange energies
are defined in the same way as the atomic
equivalents: J
AB=–2J
≠Ø
AB+J
≠≠
AB+J
ØØ
AB, with
A, BŒ{Fe, Co, Al} and the J
≠Ø
ABetc. being
proportional to the corresponding ex-
change integrals. The energy parameters
are given in Table 8-10.
The phase diagram of the b.c.c. Fe–Co
system calculated with the energy parame-
ters given in Table 8-10 is shown in Fig. 
8-34. All transitions are of second order
and have been obtained with the second
Hessian method. The calculation without
magnetic interactions is also shown. In this
non-magnetic case the critical temperature
of the B2/A2 transition is lower than in the
ferromagnetic case. The magnetic interac-
tions strengthen the atomic ordering in this
instance.
In the Co–Al system, the b.c.c. phase is
stable only around the equiatomic compo-
sition, and there the B2 structure is stable
570 8 Atomic Ordering
Figure 8-33.Calculated isothermal section of the
b.c.c. Fe–Ti–Al system at 1073 K using the energy
parameters given in Table 8-9. Second-order transi-
tions are shown as broken lines. The enlarged part of
the section shows the good agreement obtained be-
tween experiments and calculation (Ohnuma et al.,
1998b).
Table 8-10.Atomic and magnetic exchange energy
parameters in units of k
BK (Colinet et al., 1993).
A–B W
AB
(1)W
AB
(2) J
AA
(1)J
BB
(2)J
AB
(1)
Fe–Co 500 0 –163 –218 –24
Fe–Al 1680 740 –163 0 –38
Co–Al 3600 1500 –218 0 –38www.iran-mavad.com 
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622. 8.6 Application to Real Systems 571
up to the melting point. The D0
3structure
has not been observed within the b.c.c.
phase field. This imposes an upper bound
as to the value of W
(2)
CoAl
that controls the
value of the critical temperature D0
3/B2.
Based on these considerations and on ex-
perimental values for the critical tempera-
ture B2/A2 in some ternary Fe–Co–Al al-
loys, the values in Table 8-10 have been
derived by Ackermann (1988).
The b.c.c. phase diagram calculated in
the tetrahedron approximation (Colinet et
al., 1993) is shown in Fig. 8-35, super-
posed to the stable diagram. This includes
the equilibria with the liquid phase, the
f.c.c. phase, and intermetallic compounds.
The B2 phase exists up to very high tem-
peratures in a metastable state, far within
the liquid phase. This tendency is consis-
tent with the trend obtained experimentally
from data on ternary alloys (Ackermann,
1988).
Figs. 8-36 and 8-37 show the isothermal
sections of the ternary system Fe–Co–Al
at 1000 K and at 600 K. The ordered re-
gions are separated by first-order and sec-
ond-order transitions. At 600 K the mag-
netic interactions become important, but
they are still small compared with the
chemical interactions. Prior to these calcu-
lations, Ackermann (1988) performed an
MC calculation of the isothermal section at
700 K using the same chemical interaction
parameters as those in Table 8-10. Her dia-
gram and the corresponding CVM diagram
agree very well (see Inden and Pitsch,
1991).
Figure 8-34.Phase diagram of the Fe–Co system.
Second-order transitions are indicated by a hachure.
Heavy lines: CVM calculation in the (irregular) tetra-
hedron approximation using the interchange energies
given in Table 8-10 (Colinet et al., 1993); lower
curve: no magnetic interactions, upper curve: with
magnetic interactions. Experiments: (˘) Masumoto
et al. (1954), (
Ñ) Eguchi et al. (1968), (¸) Oyedele
and Collins (1977).
Figure 8-35.Phase diagram of the Co–Al system.
Second-order transitions are indicated by a hachure. Heavy lines: CVM calculation in the (irregular) tetra- hedron approximation using the interchange energies in Table 8-10 (Colinet et al., 1993). Light lines: Phase diagram according to Hansen and Anderko (1958).www.iran-mavad.com
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623. Ackermann (1988) also analyzed the
Fe–Co–Al system experimentally. Here
only one comparison will be shown. Fig. 
8-38 shows the calculated and experimen-
tal miscibility gap in a vertical section
which happened to be quasibinary (the tie-
lines are within the section).
The agreement between experiment and
calculation is surprisingly good, though not
really satisfactory. The calculated miscibil-
ity gap ends with a “horn” at the tricritical
point. This is not obtained in the experi-
ments. However, we should recall that the
two coexisting phases A2 and B2 are 
coherent with different lattice parameters.
It has been shown by Williams (1980,
1984) and Cahn and Larché (1984) that the
boundary of so-called “coherent” miscibil-
ity gaps (with elastic contributions) is
lower than the so-called “incoherent” gaps
(no elastic contributions). Therefore, elas-
tic effects due to the precipitation of coher-
ent particles depress the phase boundary.
This is only one of the effects to be taken
into account. A second one is due to the 
lattice and local relaxations within both of
these phases, as already pointed out in Sec.
8.6.2.1 for Fe–Al. The arguments present-
ed for Fe–Al are also valid for the
Fe–Co–Al system with Fe and Co being
very similar. These relaxation effects tend
to shift the miscibility gap to higher tem-
peratures, opposite to the first effect. A full
calculation is needed to evaluate the com-
bined result.
Simultaneously with the work by Acker-
mann (1988) the miscibility gap was ex-
perimentally observed by Miyazaki et al.
(1987) using transmission electron micros-
copy (TEM). These authors, however,
speculated that the miscibility gap was due
to magnetic effects and assumed the tie-
lines to be oriented towards the Fe–Co 
binary system, i.e. perpendicular to those
shown in Fig. 8-37. The contradiction be-
572 8 Atomic Ordering
Figure 8-36.Isothermal section of the phase equi-
libria of ternary b.c.c. Fe–Co–Al alloy at
T= 1000 K, calculated with the CVM in the (irregu-
lar) tetrahedron approximation using the energy pa-
rameters given in Table 8-10 (Colinet et al., 1993).
Second-order transitions are indicated by a hachure.
Figure 8-37.Isothermal section of the phase equi-
libria of ternary b.c.c. Fe–Co–Al alloys at T= 700 K, calculated with the CVM in the (irregular)
tetrahedron approximation using the energy parame- ters given in Table 8-10 (Colinet et al., 1993). Sec- ond-order transitions are indicated by a hachure. At this temperature ferromagnetic (fm) and paramag- netic (pm) states have to be distinguished.www.iran-mavad.com 
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624. 8.6 Application to Real Systems 573
tween the results of Miyazaki et al. and
those from the CVM and MC pointed out
by Inden and Pitsch (1991) was removed
by the TEM–EDS analysis of Kozakai and
Miyazaki (1994) confirming the calculated
direction of the tie-lines.
8.6.4 H.C.P. Cd–Mg
The Cd–Mg system is a prototype order-
ing system for h.c.p. alloys in the same way
as Cu–Au plays this role for f.c.c. alloys.
This system has been studied using first-
principles methods by Asta et al. (1993).
The configurational energy was evaluated
by a cluster expansion up to seven-point
clusters, as required for a treatment with
the octahedron–tetrahedron CVM. Lattice
relaxation and vibrational energy were
taken into account, but no local relaxations.
The lattice relaxation cluster expansion
was based on experimental results for dis-
ordered alloys, and the other ECIs were ob-
tained from total-energy calculations on or-
dered phases. In total 32 energy terms were
used. The ECIs were then used in the tetra-
hedron–octahedron CVM to calculate the
phase diagram, which is shown in Fig. 8-
39a. The topology of the diagram is correct
but it does not have the accuracy necessary
for technological purposes.
Zhang et al. (2000) tried the CSA in the
tetrahedron approximation. Because the
c/aratio is not ideal (f.c.c. and h.c.p. lat-
tices become equivalent in the ideal case),
two irregular tetrahedra should be consid-
ered (Onodera et al., 1994). Zhang et al.
(2000) tried the irregular tetrahedra, but
found that the regular tetrahedron CSA
gave an equally good description. The
value of 
gwas varied in such a way that the
two congruent maxima which appear at the
compositions A
3B and AB
3for g= 1.42 
(valid for the ideal h.c.p. and f.c.c., see Sec.
8.5.5) moved towards the mid-composition
until, at 
g= 1.8, the maxima disappeared,
just as observed in the Cd–Mg system.
Starting with energies taken from Asta et
al. (1993), which proved to be excellent,
only slight changes were applied in order
to get an optimum description of all the
properties that can be checked with avail-
Figure 8-38.Vertical section of
the phase diagram showing the
ternary miscibility gap as a func-
tion of composition according 
to experiment and MC (Acker-
mann, 1988), (a) Experiments
(
¯) miscibility gap, (É) critical
temperature of lro. (b) Calcula-
tion with the MC method: (
¯)
miscibility gap and critical tem-
perature of lro.www.iran-mavad.com
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625. able experimental data: the phase diagram,
the enthalpy of mixing of the disordered
state, the enthalpies of formation of or-
dered phases and the chemical potentials of
Cd and Mg. The cluster energies used were
(the values of Asta et al. are given in brack-
ets):
Cd
3Mg: –9.3 (–10.3); CdMg: –13.7 (–13.5);
CdMg
3: –9.95 (–9.5) in kJ/(mol of atoms).
The calculated phase diagram is shown
in Fig. 8-39b. The agreement between cal-
culation and experiment is very good, not
only for the phase diagram but also for all
the other properties mentioned above.
8.6.5 Concluding Remarks
The examples treated above represent
the large group of real systems that can be
analyzed using these techniques. Similar
findings were obtained for oxide systems,
e.g., by Burton (1984, 1985), Burton and
Kikuchi (1984), Kikuchi and Burton (1988),
Burton and Cohen (1995), Tepesch et al.
(1995, 1996), Kohan and Ceder (1996).
At present, it can be concluded that the
CVM and MC techniques supply a treat-
ment of sufficient sophistication to cor-
rectly handle the statistical aspects of the
equilibria in solid solutions.
Much progress has been made in the
field of first-principles calculations of total
energies, including lattice and local relaxa-
tions, and sometimes including excitations.
From this the energy parameters of the sta-
tistical models can be calculated. The re-
sults for binary systems are numerous, but
the field of multicomponent systems is still
to be explored. With increasing numbers of
components the cluster size also has to be
increased.
For the solution of metallurgical prob-
lems, the phase equilibria between all
phases have to be considered. The order–
disorder equilibria within a given crystal
structure are only one part of this task.
Other phases, such as the liquid phase or
intermetallic compounds, have to be in-
cluded. These aspects have been discussed
at recent workshops on the thermodynamic
modeling of solutions and alloys (e.g. Cac-
ciamani et al., 1997).
574 8 Atomic Ordering
Figure 8-39.Calculated phase diagrams of the hexagonal Cd–Mg system. a) Phase diagram calculated from 
“first principles” (redrawn from Asta et al., 1993). b) Phase diagram calculated with the CSA using almost the
same parameters as in a) and a value 
g= 1.8. Experimental data from Frantz and Gantois (1971).www.iran-mavad.com 
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626. 8.7 Appendix 575
8.7 Appendix
Table 8-11.Superstructures of the f.c.c. lattice.
1
Designation
2
Spacegroup Basis
3
Equivalent positions Occupation
Positions
A1 (Cu) Fm3m a= a
0{1 0 0} (0 0 0; 1/2 1/2 0; 1/2 0 1/2; 0 1/2 1/2) A/B
A
5BB 2/m a= a
0/2 [1 1
–
2] (0 0 0; 1/2 0 1/2)+
Fig. 8-40 b= a
0/2 [2 2
–
2] (0 0 0) 2 B
c= a
0/2 [3 3 0] (0 1/2 0) 2 A
(0 0 ±1/3) 4 A
(0 1/2 ±1/3) 4 A
D1a (Ni
4Mo) I4/m a
1= a
0/2 [3 1
–
0] (0 0 0; 1/2 0 1/2)+
Fig. 8-41 a
2= a
0/2 [1 3 0] (0 0 0) 2 B
c= a
0/2 [0 0 2] ( x y 0; x
–
y
–
0; y
–
x0; yx
–
0) 8 A
x= a
0÷
ææ
2/5y= a
0/ ÷
æ
10 
L1
2(Cu
3Au) Pm3m a= a
0{1 0 0} ––
(0 0 0) B
(1/2 1/2 0; 1/2 0 1/2; 0 1/2 1/2) 3 A
–– P4/mmm a
1= a
0[1 0 0] ––
a
2= a
0[0 1 0] (0 0 0) B
c= a
0[0 0 1] (1/2 1/2 0) A B
(1/2 0 1/2; 0 1/2 1/2) A
D0
22(Ti
3Al) I4/mmm a
1= a
0/2 [0 1 0] (0 0 0; 1/2 1/2 1/2)+
Fig. 8-42 a
2= a
0/2 [0 0 1] (0 0 0) 2 B
c= a
0[2 0 0] (0 0 1/2) 2 A
(0 1/2 1/4; 1/2 0 1/4) 4 A
A
3B Amm2 a= a
0/2 [0 2 2] (0 0 0; 1/2 0 1/2)+
b= a
0/2 [0 1
–
1] (0 0 0) 2 B
c= a
0/2 [0 0 2] (0 0 1/2) 2 A
(1/4 1/2 1/4; 1/4 1/2 3/4) 4 A
A
2B (Pt
2Mo) Immm a= a
0/2 [1 1
–
0] (0 0 0; 1/2 1/2 1/2)+
Fig. 8-43 b= a
0[1 0 0] (0 0 0) 2 B
c= a
0/2 [3 3 0] (0 0 ±1/3) 4 A
L1
0(CuAu) P4/mmm a
1= a
0/2 [1 1 0] ––
a
2= a
0/2 [1 1
–
0] (0 0 0) B
c= a
0[0 0 1] (1/2 1/2 1/2) A
L1
1(CuPt) R 3
–
m a
1= a
0/2 [2 1 1] ––
Fig. 8-44 a
2= a
0/2 [1 2 1] (0 0 0) B
a
3= a
0/2 [1 1 2] (1/2 1/2 1/2) A
A
2B
2 I41/amd a
1= a
0[0 1 0] (0 0 0; 1/2 1/2 1/2)+
Fig. 8-45 a
2= a
0[0 0 1] (0 0 0; 0 1/2 1/4) B
c= a
0[2 0 0] (0 0 1/2; 0 1/2 3/4) A
1 Figures 8-40 to 8-45 represent the original f.c.c. unit cell and the unit cells of the superstructures; 2 “Struk-
turbericht” designation; 3 In terms of vectors of the f.c.c. structure.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

627. 576 8 Atomic Ordering
Figure 8-40.Original f.c.c. unit cell and unit cell of
the superstructure A
5B (B2/m), indicated by bold
lines.
Figure 8-43.Original f.c.c. unit cell and unit cell of
the superstructure A
2B (Pt
2Mo), indicated by bold
lines.
Figure 8-41.Original f.c.c. unit cell and unit cell of
the superstructure D1a (Ni
4Mo), indicated by bold
lines.
Figure 8-42.Original f.c.c. unit cell and unit cell of
the superstructure D0
22(Ti
3Al), indicated by bold
lines.
Figure 8-45.Original f.c.c. unit cell and unit cell of
the superstructure A
2B
2(I4/amd), indicated by bold
lines.
Figure 8-44.Original f.c.c. unit cell and unit cell of
the superstructure L1
1(CuPt), indicated by bold
lines.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

628. 8.7 Appendix 577
Table 8-12.Superstructures of the b.c.c. lattice.
Designation
1
Spacegroup Basis
2
Equivalent positions Occupation
Positions
A2 (Fe) Fd3c a= a
0{1 0 0} –– A/B
D0
3(Fe
3Al) Fm3m a
1= a
0[2 0 0] ––
a
2= a
0[0 2 0] (0 0 0) 4 B
a
3= a
0[0 0 2] (1/2 1/2 1/2) 4 A
(1/4 1/4 1/4; 3/4 3/4 3/4) 8 A
F 4
–
3 m a
1= a
0[2 0 0] ––
a
2= a
0[0 2 0] (0 0 0) 4 B
a
3= a
0[0 0 2] (1/2 1/2 1/2) 4 A
(1/4 1/4 1/4) 4 A/B
(3/4 3/4 3/4) 4 A
B2 (CsCl) Pm3m a= a
0{1 0 0} ––
(0 0 0) A
(1/2 1/2 1/2) B
B32 (NaTl) Fd3m a
1= a
0[2 0 0] (0 0 0; 0 1/2 1/2; 1/2 0 1/2; 0 1/2 1/2)+
a
2= a
0[0 2 0] (0 0 0; 1/4 1/4 1/4) 8 A
a
3= a
0[0 0 2] (1/2 1/2 1/2; 3/4 3/4 3/4) 8 B
1 “Strukturbericht” designation; 2 In terms of vectors of the b.c.c. structure.
Table 8-13.Superstructures of the hexagonal lattice.
Designation
1
Spacegroup Basis
2
Equivalent positions Occupation
Positions
A3 F6
3/mmc a= a
0[1 0 0] ––
b= a
0[0 1 0] (1/3 2/3 3/4) 2 (A/B)
c= a
0[0 0 1]
D0
19(A
3B) P63/mmc a= a
0[2 0 0] ––
b= a
0[0 2 0] 6 h (5/6 2/3 1/4) 6 A
c= c
0[1 0 0] 2 c (1/3 2/3 1/4) 2 B
B19 Pmma a= c
0[0 0 1
–
]– –
b= a
0[0 1 0] 2 f (1/4 1/2 5/6) 2 A
c= a
0[2 1 0] 2 e (1/4 0 1/3) 2 B
A
2BP 2
1/m a= a
0[1 1
–
0] 2 e (1/2 1/6 1/4) A
b= a
0[1 2 0] 2 e (1/6 1/2 1/4) A
c= c
0[0 0 1] 2 e (5/6 5/6 1/4) B
A
2Bp 6m a= a
0[1 1
–
] 2 b (1/3 2/3) A
(2-dim.) b= a
0[1 2] 1 a (0 0) B
1 “Strukturbericht” designation; 2 In terms of vectors of the h.c.p. structure.www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

629. 8.8 Acknowledgements
In writing this chapter the author has
benefited from the work of past students,
Claudio G. Schön, Sabine Crusius and
Helen Ackermann, and from intense co-op-
eration with colleagues, Ryoichi Kikuchi,
Catherine Colinet and Christophe Bichara.
This is gratefully acknowledged. The con-
cept of multicomponent correlation func-
tions was worked out in co-operation with
Alphonse Finel when writing the version
for the first edition of this volume (Inden
and Pitsch, 1991). Thanks are due to Alan
Oates for very helpful and enlightening
discussions.
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633. 9 Diffusionless Transformations
Luc Delaey
Departement Metaalkunde en Toegepaste Materiaalkunde,
Katholieke Universiteit Leuven, Heverlee-Leuven, Belgium
List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 585
9.1 Introduction................................ 587
9.2 Classification and Definitions....................... 590
9.3 General Aspects of the Transformation................. 593
9.3.1 Structural Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
9.3.2 Pre-transformation State . . . . . . . . . . . . . . . . . . . . . . . . . . 597
9.3.3 Transformation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 599
9.3.4 Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
9.3.5 Shape Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
9.3.6 Transformation Thermodynamics and Kinetics . . . . . . . . . . . . . . . 604
9.4 Shuffle Transformations.......................... 607
9.4.1 Ferroic Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
9.4.2 Omega Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
9.5 Dilatation-Dominant Transformations.................. 610
9.6 Quasi-Martensitic Transformations................... 611
9.7 Shear Transformations.......................... 613
9.8 Martensitic Transformations....................... 615
9.8.1 Crystallography of the Martensitic Transformation . . . . . . . . . . . . . 615
9.8.1.1 Shape Deformation and Habit Plane . . . . . . . . . . . . . . . . . . . . 615
9.8.1.2 Orientation Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . 616
9.8.1.3 Morphology, Microstructure and Substructure . . . . . . . . . . . . . . . 618
9.8.1.4 Crystallographic Phenomenological Theory . . . . . . . . . . . . . . . . 620
9.8.1.5 Structure of the Habit Plane . . . . . . . . . . . . . . . . . . . . . . . . . 623
9.8.2 Thermodynamics and Kinetics of the Martensitic Transformation . . . . . 624
9.8.2.1 Critical Driving Force and Transformation Temperatures . . . . . . . . . 624
9.8.2.2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
9.8.2.3 Growth and Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
9.8.2.4 Transformation Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . 634
9.9 Materials.................................. 634
9.9.1 Metallic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634
9.9.1.1 Ferrous Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
9.9.1.2 Non-Ferrous Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
9.9.2 Non-Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
9.10 Special Properties and Applications................... 641
9.10.1 Hardening of Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
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634. 9.10.2 The Shape-Memory Effect . . . . . . . . . . . . . . . . . . . . . . . . . 641
9.10.3 High Damping Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
9.10.4 TRIP Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
9.11 Recent Progress in the Understanding of Martensitic Transformations649
9.12 Acknowledgements............................. 651
9.13 References................................. 652
584 9 Diffusionless Transformationswww.iran-mavad.com
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635. List of Symbols and Abbreviations 585
List of Symbols and Abbreviations
a length
A factor (containing elastic terms)
a, b, c constants
A
d retransformation temperature (deformation induced)
A
j amplitude of perturbation with polarization j
A
s starting temperature for austenite formation
B pure strains associated with lattice correspondence
C lattice correspondence
C cubic sequence
C¢ elastic shear constant
c* size of critical nucleus
C
ij eigenvalues of elasticity tensor
c/a axial ratio
c/r thickness/radius ratio of nucleus
e order parameter
E electric field
e
1, e
2, e
3 principal strains
E
e strain energy
F applied force
DG difference in chemical Gibbs energy
DG* Gibbs energy of nucleation
Dg
s surface Gibbs energy per unit volume
G* elastic state function
G
a
, G
b
, G
g
Gibbs energy of phases a, b, g
G
c, G
chem chemical Gibbs energy
G
elast elastic Gibbs energy
G
surf surface Gibbs energy
G
tot total Gibbs energy
G
d defect Gibbs energy
G
i interaction Gibbs energy
H magnetic field
H* elastic state function
DH, DH* enthalpy change
l molar length
M
d deformation-induced martensitic temperature
M
f martensite finishing temperature
M
s martensite starting temperature
P inhomogeneous lattice-invariant deformation
q wave vector
r radius of a plate
r lattice vector
R rigid body rotation
R rhombohedral sequencewww.iran-mavad.com
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636. r* size of critical nucleus
S strain matrix
DS entropy change
T temperature
T
0 equilibrium temperature
T
c critical transition temperature
T
N Neel temperature
T
TW temperature at which twins appear
v volume of a plate
V
m molar volume
x atomic fraction of elements
x, y lattice vectors
a name of phase
b name of phase
g name of phase
G interfacial energy
d
0 shear strain
e name of phase
e
0 strain associated with the transformation
j surface to volume ratio
s stress
s
a applied stress
ASM American Society for Materials
b.c.c. body-centered cubic
b.c.t. body-centered tetragonal
f.c.c. face-centered cubic
f.c.t. face-centered tetragonal
G–T Greninger–Troiano
h.c.p. hexagonal close packed
HIDAMETS high-damping metals
HP habit plane
HRTEM high-resolution transmission electron microscopy
IPS invariant plane strain
K–S Kurdjurnov–Sachs
LOM light optical microscopy
N–W Nishiyama–Wassermann
PTFE polytetrafluoroethylene
SMA shape-memory alloys
SME shape-memory effect
TRIP transformation-induced plasticity
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637. 9.1 Introduction 587
9.1 Introduction
Diffusionless solid-state phase transfor-
mations, as the name suggests, do not re-
quire long-range diffusion during the phase
change; only small atomic movements over
usually less than the interatomic distances
are needed. The atoms maintain their rela-
tive relationships during the phase change.
Diffusionless phase transformations there-
fore show characteristics (such as crystal-
lographic, thermodynamic, kinetic and mi-
crostructural) very different from those of
diffusive phase transformations.
Martensitic transformations, because
some of the properties associated with them
sometimes lead to specialized applications,
are considered to be an extreme class of
diffusionless phase transformation and we
therefore in this chapter concentrate on
martensite. Because of the similarity of
some of the transformation characteristics,
a number of other diffusionless solid-state
phase transformations have sometimes
been designated erroneously as marten-
sitic. To avoid misinterpretations, Cohen
et al. (1979) proposed a classification
scheme that identifies broad categoris of
displacive transformations showing fea-
tures in common with martensitic transfor-
mations, but distinct from them. Their clas-
sification scheme, reproduced in Fig. 9-1,
will largely be followed here. Martensitic
transformations are here only a subclass of
the broader class of displacive/diffusion-
less phase transformations.
The classification proposed by Cohen et
al. (1979) is discussed first, and subsequent
sections deal with general aspects of the
crystallography, thermodynamics and ki-
netics of the different displacive transfor-
mations. Although it is not the purpose to
give full details of all materials that exhibit
this type of transformation, the most im-
portant material systems in which such
transformations have been observed are
presented.
A martensitic transformation can be de-
tected by a number of techniques; some are
in situmethods, whereas others are step-
by-step measurements. The results are usu-
ally plotted as a change in a physical prop-
erty versus temperature (see the schematic
representation in Fig. 9-2), from which the
transformation temperatures can be deter-
mined. Some of these plots can be trans-
lated into the volume fraction of martensite
formed versus the temperature. Such
curves allow us to determine the transfor-
mation temperatures as indicated in these
figures.
In situdetection becomes limited if the
transformation temperature is above room
temperature, and dilatometry then seems to
be the most appropriate technique provided
that quenching – which is needed to avoid
alterations in the sample due to diffusion –
is possible inside the dilatometer. There are
far more ways of following the transforma-
tion when the transformation temperature
is below room temperature – preparing the
sample and carrying out the measurements
can take some time and at room tempera-
ture diffusion is then almost negligible.
During slow cooling after water quench-
ing, the techniques frequently used include
dilatometry, electrical resistivity and mag-
netic measurements, calorimetry, in situ
microscopy, acoustic emission, elastic and
internal friction measurements, positron
annihilation, and Mössbauer spectroscopy.
Some of the less common techniques used
to study martensitic transformations were
reviewed by Fujita (1982).
If crystallographic information is re-
quired, X-ray and electron or neutron dif-
fraction are used. X-ray diffraction meas-
urements by Fink and Cambell in 1926 (lat-
tice parameter of C-steel martensite), by
Kurdjurnov and Sachs in 1930, by Ni-www.iran-mavad.com
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638. shiyama in 1934 and by Greninger and
Troiano in 1949 (orientation relationship
between austenite – the parent phase – and
martensite) represent breakthroughs in the
study of martensitic transformations.
The most frequently used techniques
will now be briefly discussed and illus-
trated.
If a sample is polished into the marten-
site (= parent phase), a surface relief ap-
pears. An edge-on section of such a sample
is shown in Fig. 9-2a (see Hsu, 1980). The
origin of the surface relief is indicated by
the white arrows. Owing to the macro-
scopic martensite shear (the two thinner ar-
rows), a surface relief is obtained. This is
588 9 Diffusionless Transformations
Figure 9-1.Classification scheme for the displacive/diffusionless phase transformations as proposed by Cohen
et al. (1979).www.iran-mavad.com
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639. 9.1 Introduction 589
Figure 9-2.Some examples of how to see or measure the presence and growth of martensite (see text for de-
tails). www.iran-mavad.com
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640. explained further in Sec. 9.8.1.1 (Fig.
9-19).
The electrical resistance shows at the
transformation temperatures a deviation
from linearity versus temperature. This is
shown in Fig. 9-2b for the martensitic
transformation in an Fe–Ni alloy and a
Au–Cd alloy. The difference in the resis-
tance ratio for the two different materials is
obvious and remarkable (see Otsuka and
Wayman, 1977). Measuring the electrical
resistance while cooling or heating a sam-
ple is a very convenient and relatively easy
and accurate technique of determining the
transformation temperatures M
s, M
f, A
s
and A
f.
The heat exchanged with the surround-
ings is becoming a more popular method of
determining the transformation tempera-
tures. An example is shown in Fig. 9-2c
(Nakanishi et al., 1993). A DSC (differen-
tial scanning calorimetry) curve allows any
particular behavior of the martensitic sam-
ple to be detected (for example, effects oc-
curring during heat treatments and/or def-
ormation steps).
A number of martensitic transformations
and materials are characterized by a so-
called shape-memory effect (see Sec.
9.11.2). Figure 9-2d (courtesy of Memory
Europe) shows the displacement of a
spring made of a NiTi shape-memory alloy.
The spring controls a small valve in a cof-
fee-making machine. At the temperature A
s
the hot water starts to drop onto the coffee
powder. This valve is completely open as
soon as the temperature A
fis reached. The
temperature range between A
sand A
fseems
to be most suitable for making the best cup
of coffee. On cooling, the valve closes
again. The “displacement–temperature”
curve measured on cooling does not coin-
cide with the heating curve.
During a martensitic transformation, not
only is the shape of the sample changed but
also the specific volume, which allows the
transformation temperatures to be deter-
mined by dilatometry (Fig. 9-2e, from
Yang and Wayman, 1993).
Changes in mechanical properties are
also measured while the sample is trans-
forming. The Young’s modulus exhibits a
dip between the two transformation tem-
peratures M
sand M
f, as clearly visible in
Fig. 9-2f for four different alloys (see Su-
gimoto and Nakaniwa, 2000).
9.2 Classification and
Definitions
A structrual change in the solid state is
termed “displacive” if it occurs by coordi-
nated shifts of individual atoms or groups
of atoms in organized ways relative to their
neighbors. In general, this type of transfor-
mation can be described as a combination
of “homogeneous lattice-distortive strain”
and “shuffles”.
A lattice-distortive deformation is a ho-
mogeneous strain that transforms one lat-
tice into another; examples are shown in
Fig. 9-3. The homogeneous strain can be
represented by a matrix according to
y=Sx (9-1)
where the strain Sdeforms the lattice vec-
tor xinto a lattice vector y. This strain
is homogeneous because it transforms
straight lines into other straight lines. A
spherical body of the parent phase will thus
be transformed into another sphere or into
an ellipsoidal body. The actual shape of the
ellipsoid depends on the deformation in the
three principal directions. If a spherical
body is completely embedded inside the
matrix phase and is undergoing the strain
S, the volume and shape change associated
with this deformation will cause elastic
and, sometimes, plastic strains in the parent
590 9 Diffusionless Transformationswww.iran-mavad.com
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641. 9.2 Classification and Definitions 591
and/or product phases. The lattice-distor-
tive displacements therefore give rise to
elastic strain energy. In addition, an inter-
face separating the phases is created, gen-
erating an interfacial energy. As is obvious
from Fig. 9-1, the relative values of these
energies play an important role in the clas-
sification scheme.
A shuffle is a coordinated movement of
atoms that produces, in itself, no lattice-
distortive deformations but alters only the
symmetry or structure of the crystal; a
sphere before the transformation remains
the same sphere after the transformation.
Shuffle deformations produce, in the ideal
case, no strain energy and thus only interfa-
cial energy. Two examples of the shuffle
displacement are given in Fig. 9-4. Shuffle
deformations can be expressed by “lattice
Figure 9-3.Examples of the lattice-distortive defor-
mations of a cubic lattice: (1) a dilatation in the three
principal directions transforms the lattice into an-
other cubic lattice with larger lattice parameters; (2)
a shear along the (001) plane leads to a monoclinic
lattice, and (3) an extension along the [001] axis
combined with a contraction along the [100] and
[010] axis results in an orthorhombic lattice.
Figure 9-4.Examples of shuffle displacements in:
(a1) strontium titanate; Ωoxygen, ≤strontium, ≤
ΩΩΩti-
tanium; (a2) the displacement of some of the oxygen atoms can be represented by an alternating clockwise and anti-clockwise rotation around the titanium at- oms; and (b) the (111) planes in a b(b.c.c.) lattice
and the collapsed (0001) planes in the hexagonal w
lattice (Sikka et al., 1982).www.iran-mavad.com
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642. wave modulations” as
Dx=A
j(q) exp (iq·r) (9-2)
where ris a lattice vector, q is the wave
vector giving the direction and inverse
wavelength of the modulation, Ais the am-
plitude of the perturbation, and jdenotes
the polarization of the wave. An alternative
description is represented by relative dis-
placements of the various atomic sub-lat-
tices that specify the structures of the two
phases in terms of corresponding unit cells,
which are not necessarily primitive.
Cohen et al. (1979) subdivided the dis-
placive transformations into two main
groups, according to the relative contribu-
tion of the two above-mentioned atom dis-
placements and hence the ratio of the inter-
facial/strain energy. In this context, they
distinguish between “shuffle transforma-
tions” and “lattice-distortive transforma-
tions”. Since the latter give rise to elastic
strain energy and the former only to inter-
facial energy, major differences are found
in the kinetic and morphological aspects of
the transformation, which justifies the dis-
tinction. Shuffle transformations are not
necessarily pure; small distortive deforma-
tions may additionally occur. They there-
fore also include those transformations in-
volving dilatational displacements, in addi-
tion to the pure shuffle displacements, pro-
vided that they are small enough not to al-
ter significantly the kinetics and morphol-
ogy of the transformation.
The lattice-distortive transformations
themselves are subdivided according to the
relative magnitudes of the two components
of the homogeneous lattice deformation,
i.e., the dilatational and the deviatoric
(shear) components (see Fig. 9-3). The in-
itial and the isotropically dilated spheres
have no intersection and it is therefore not
possible to find a vector whose length has
not been changed by the transformation.
On the other hand, the ellipsoid obtained
after a pure shear intersects the original
sphere; hence a set of vectors exist, whose
lengths remain unchanged. Such a defor-
mation is said to be characterized by an
“undistorted line“. An undistorted line can
only result from a homogeneous lattice
deformation if the deviatoric or shear com-
ponent sufficiently exceeds the dilatational
component.
Cohen et at. (1979) thus consider a trans-
formation as deviatoric-dominant if an in-
variant line exists. A further subdivision
was made between phase transformations
with and without an invariant line, or
between “dilatation-dominant” and “devi-
atoric-dominant” transformations.
If the magnitude of the lattice-distortive
displacements is large in relation to that of
the lattice vibrational displacements, high
elastic strain energies are involved. How-
ever, if they are comparable, the strain
energies will be small and hence will not
dominate the kinetics and morphology of
the transformation. In the latter case we
deal with “quasi-martensitic transforma-
tions”. The former, i.e., the “martensitic
transformations”, are therefore those dis-
placive or diffusionless phase transforma-
tions where the lattice-distortive displace-
ments are large enough to dominate the ki-
netics and the morphology of the transfor-
mation. “Martensite” is the name now
given to the product phase resulting from a
martensitic transformation. Because of the
volume change and the strain energy in-
volved with the transformation, martensitic
transformation requires heterogeneous nu-
cleation and passes through a two-phase
mixture of parent and product; it is a first-
order diffusionless phase transformation.
Consequently, the forward and reverse
transformations are accompanied by an ex-
othermic and endothermic heat effect, re-
spectively, and forward and reverse trans-
592 9 Diffusionless Transformationswww.iran-mavad.com
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643. 9.3 General Aspects of the Transformation 593
formation paths are separated by a hystere-
sis.
Among the various diffusionless phase
transformations which exist in solid-to-
solid phase transformations, martensitic
transformations have received much atten-
tion in the past. Historically, the term “mar-
tensite” was suggested by Osmond in 1895,
in honor of the well-known German metal-
lurgist Adolph Martens, as the name for the
hard product obtained during the quench-
ing of carbon steels. It was found that the
transformation to martensite in steel was
associated with a number of distinctive
characteristic structural and microstructu-
ral features. During the last few decades it
was recognized that martensite also forms
in numerous other materials, such as super-
conductors, non-ferrous copper-based al-
loys, zirconia, which recently became a
popular research subject for ceramists,
physicists, chemists and polymeric and bi-
ological scientists.
Martensitic transformations have been
the subject of a series of international con-
ferences held in various places throughout
the world. A list is given at the end of this
chapter.
This increasing interest not only has aca-
demic origins but can to a large extent also
be attributed to a number of industrial ap-
plications such as maraging, TRIP (trans-
formation-induced plasticity) and dual-
phase steels, applications involving the
shape-memory effect, the high damping ca-
pacity, and the achievement of transforma-
tion toughening in ceramics.
It may also be of interest to draw atten-
tion here to the “massive phase transforma-
tions”. Although this type of phase trans-
formation, which occurs upon fast cooling,
is composition-invariant and the transfor-
mation interface has a relatively rapid
movement, it does not fall into the present
category. Massalski (1984) defines mas-
sive transformation as a non-martensitic
composition-invariant reaction involving
diffusion at the interfaces (see also the
chapter by Purdy and Bréchet (2001)).
“Bainite transformations” are also not
treated in this chapter, although they do
show some martensitic characteristics, but
combined with diffusional processes. For
further information, the reader is referred
to Aaronson and Reynolds (1988) for an in-
troductory review and to Krauss (1992).
9.3 General Aspects
of the Transformation
The various diffusionless phase transfor-
mations have a number of features in com-
mon, such as the crystallographic aspects
of the structural changes, the pretransfor-
mation state, the transformation mecha-
nisms, the microstructure and the shape
changes that result from the transforma-
tion, and the thermodynamic and kinetic
aspects. The more general aspects are
treated in the following section before dis-
cussing separately each subclass of trans-
formation.
9.3.1 Structural Relations
This section is concerned with some
crystallographic aspects of the structural
changes. It is always useful to first deter-
mine a unique relationship – a lattice corre-
spondence (C) – between any vector in the
initial lattice and the vector that it becomes
in the product lattice. A lattice correspon-
dence thus defines a structural unit in the
parent phase that, under the action of a ho-
mogeneous deformation, is transformed
into a unit of the product phase. Such a cor-
respondence therefore tells us which vec-
tors, planes and unit cells of the product
phase are derived from particular vectors,www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

644. planes and cells of the parent phase, with-
out regard to their mutual orientation. For
every structural change there exist many
ways of producing a lattice correspon-
dence; the one that involves minimal
atomic displacements and which reflects
the experimentally observed orientation re-
lationships most closely should be se-
lected.
The actual relationship between labelled
vectors, planes, etc., before and after trans-
formation (including their mutual orienta-
tion) is given by the lattice deformation.
Mathematically this lattice deformation is
factorized into a pure strain and a pure
rotation, so the correspondence indicates
what is the pure strain. Knowing the princi-
pal axes of the strain ellipsoid, the direc-
tions of undistorted lines, if any exist, can
then easily be found. In 1924, Bain pro-
posed such a lattice correspondence for the
f.c.c.-to-b.c.c. (or b.c.t.) transformation in
iron alloys; it is referred to in the literature
as the prototype Bain correspondence and/
or Bain strain. Since then, lattice corre-
spondence values (C) and their associated
pure strains (B) have been published for a
number of other structural changes; some
examples are given in Fig. 9-5.
In the original Bain strain, a tetragonal
cell is delineated into two adjacent f.c.c.
unit cells. Then, it is contracted along zby
about 20% and is expanded along x¢and y¢
by about 12%. In another example, the
transformation from a NaCl-type structure
into a CsCl-type structure, a contraction of
40% along the [111] body diagonal and a
19% isotropic expansion in the perpendicu-
lar (111) plane is needed; the volume
change is about 17%.
Homogeneous strains alone, however, do
not always describe the structural transfor-
mation completely. Additional shuffles
may be needed to obtain the exact atom ar-
rangements inside the deformed unit cell.
A special situation arises for some mate-
rials, when the structural change can be
achieved by a pure deformation that leaves
one of the principal directions unaltered.
Such a situation is found in some polymers.
It occurs, for example, in polyethylene,
which has an orthorhombic and a mono-
clinic polymorph with chains parallel to the
z-axis; these strong covalently bonded
chains are unlikely to be distorted by the
transformation; consequently, the pure def-
ormation along the z-axis (e
3= 0) is zero.
During transformation the chains are dis-
placed transversely in such a way that one
594 9 Diffusionless Transformations
Figure 9-5.Some examples of lattice correspon-
dence and homogeneous deformation (expansions
and contractions) for (a) f.c.c. to b.c.c. or b.c.t. (after
Bain, 1924) and (b) NaCl- to CsCl-type structures
(the Shôji–Buerger lattice deformation) (Kriven,
1982).www.iran-mavad.com
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645. 9.3 General Aspects of the Transformation 595
of the remaining principal strains becomes
positive (e
2> 0) and the other negative
(e
1> 0) (Bevis and Allan, 1974).
The strain ellipsoid for the above exam-
ple has a special shape. The cone of undis-
torted vectors of the product phase degen-
erates into a pair of planes, which rotate in
opposite directions in the pure strain.
Hence either of them may be invariant if
the total deformation of the lattice includes
a suitable rotation. Because all the vectors
in this plane are undistorted, the transfor-
mation is said to be an “invariant plane
strain (IPS)” type. The pure Bain strain is
then equivalent to a simple shear on that in-
variant plane. Because this invariant plane
is also a matching plane between the ma-
trix and the product phase and both phases
have to be present at the same time, a rigid
body rotation (R) over an angle
qis re-
quired in order to bring the product and the
parent phases in contact along that plane,
the habit plane.
A similar situation is found in structu-
rally less sophisticated systems, namely the
f.c.c. to h.c.p. transformation in cobalt.
Both phases are close packed and a simple
shear on the basal plane transforms the cu-
bic stacking into a hexagonal stacking. Be-
cause the atomic distances in the basal
plane do not change significantly during
the transformation, the plane of simple
shear is the plane of contact or the habit
plane (HP). This is the case, however, only
if there is zero volume change in the trans-
formation, i.e., in the case quoted above if
the h.c.p. phase has an ideal axial ratio of
1.633.
The situation becomes more complicated
if none of the principal strains is zero, but
of mixed sign. To achieve matching along
the plane of contact in cases where the two
phases coexist, a deformation is needed ad-
ditional to the pure Bain strain in order to
have an invariant plane. Because the final
lattice hase already been generated by the
Bain strain, this additional strain should be
a “lattice-invariant strain”. Slip and twin-
ning in the product phase or in both phases
are typical lattice-invariant strains; both
deformation modes are shown schemati-
cally in Fig. 9-6. The diffusionless phase
transformation can in this case be repre-
sented by an analog consisting of a pure
lattice strain (B), a rigid lattice rotation (R)
and an inhomogeneous lattice-invariant
deformation (P). The last factor is also as-
sociated with a shape change, which mac-
roscopically can be considered as homoge-
neous. Such a combination is typical of
martensite and is discussed in Sec. 9.8.1.4.
In cases where the lattice-invariant shear
is twinning (as opposed to faulting or slip),
type I twinning, where the twin plane orig-
inates from a mirror plane in the parent
phase, has been assumed. Otsuka (1986)
carefully analyzed for a number of systems
the possibility of a type II twinning as an
alternative inhomogeneous shear. In type II
twinning, the shear direction stems from a
two-fold symmetry direction of the parent
phase. In a table, Otsuka (1986) compiled
all the twinning modes observed in marten-
Figure 9-6.Schematic representation of (a) the ho-
mogeneous lattice deformation, (b) the inhomogene-
ous lattice-invariant deformation (slip and twinning),
and (c) the lattice rotation.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

646. site and found that most of them are type I
or compound but that type II twinning had
only recently been observed. According to
Nishida and Li (2000), five different twin-
ning modes exist in TiNi and other shape
memory alloys such as Cu-Al-Ni, Cu-Sn
etc., namely the {111} type I, the {011}
type I, the 〈011〉 type II, the (100)-com-
pound and the (001)-compound twins.
Type II twinning has recently received
much attention as a mechanism for lattice
invariant shear in some alloys. Since
type II twins have irrational twin boundar-
ies, the physical meaning of an irrational
boundary is still a controversial problem. It
has proposed that an irrational boundary
consists of rational ledges and steps, the
average being irrational. Thereafter, Hara
et al. (1998) carried out a careful study to
observe 〈111〉type II twin boundaries in a
Cu-Al-Ni alloy by HRTEM, but they could
not observe any ledges or steps. The boun-
dary is always associated with dark strain
contrast, and the lattice is continuous
through the irrational boundary. Nishida
and Li (2000) also made extensive studies
on 〈011〉type II twin boundaries in TiNi by
HRTEM, but they did not observe ledges or
steps either. Based on these experimental
results, it is thus most likely that the type II
thin boundary is irrational even on a micro-
scopic scale, and the strains at the boun-
dary are elastically relaxed with wide twin
width. To confirm this interpretation, Hara
et al. (1998) carried out computer simula-
tions by using the molecular dynamics
method. The result showed that the irra-
tional thin boundary did not show any
steps. Thus, the above interpretation for an
irrational twin boundary is justified. Ot-
suka and Ren (1999) have pointed out
again the importance of type II twinning in
the crystallographic aspects of martensite.
They also stress the role that martensite ag-
ing has on the rubber-like behaviour of
martensite, a point that has been a long-
standing unsolved problem. They showed
that the point defects play a primordial
role. The deformation mechanisms of the
cold deformation of NiTi martensite have
been thoroughly analyzed by Liu et al.
(1999a, b). They also found an interplay
between type I and type II twinning.
The cubic to tetragonal transformations,
which occur in a number of metallic and
non-metallic systems, need some special
attention. The volume change with these
transformations is sometimes very small or
even absent, and the c/aratio does not
change abruptly but progressively (Fig.
9-7); the transformation is then said to be
“continuous”. The c/aratio can be smaller
or larger than unity, depending on compo-
sition and temperature. The shape change
associated with the transformation is small
enough in many systems, especially in
those belonging to the quasi-martensites,
for elastic accommodation alone to be suf-
596 9 Diffusionless Transformations
Figure 9-7.Temperature
dependence of c/aas mea-
sured during the cubic to
tetragonal transformation;
(a) second and (b) first-or-
der phase transformation.www.iran-mavad.com
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647. 9.3 General Aspects of the Transformation 597
ficient for lattice matching. It is, however,
possible for cand ato change abruptly with
zero volume change.
Based on the crystallographic aspects
discussed above, a list of the most typical
characteristics of the diffusionless phase
transformations can be compiled (Table
9-1).
9.3.2 Pre-transformation State
Diffusionless structural changes are
achieved by atom displacements, such as
shuffles and shears. The new atomic con-
figuration is already prepared in some ma-
terial systems at temperatures above the
transition temperature. Atoms in the parent
phase then become displaced more easily
towards their positions in the new phase
because the restoring force that is felt by
the displaced atoms diminishes on cooling.
In certain cases the restoring force even
vanishes at the phase transition tempera-
ture.
Certain shuffle transformations result
from a vibrational instability of the parent
phase and are therefore called “softmode”
phase transformations. A soft mode is,
in simple terms, a vibrational mode, the
square of whose frequency tends toward
zero as the temperature approaches that of
the phase transition. The average static
atom displacements resemble the frozen-in
pattern of the vibrational displacements of
a certain vibrational mode. According to
Vallade (1982), “the crystal lattice vibra-
tions can within the harmonic approxima-
tion be separated into independent plane
waves (phonons) characterized by a set
of collective atomic displacements corre-
sponding to a well defined frequency. The
energy involved is a function of the squares
of the momentum and of the eigenfrequen-
cies of the mode. The eigenfrequencies de-
pend only on the mass of the atoms and on
the force constants. It is clear that the van-
ishing of one eigenfrequency corresponds
to the lack of restoring force for the mode:
the amplitude can then grow without any
limit and the lattice is mechanically un-
stable. Stability can be recovered only
by changing atomic equilibrium positions
which, in turn, changes the force con-
stants“.
Table 9-1.A schematic overview of some characteristics typical of the various types of diffusionless phase
transformations.
Characteristics Structural change Pure lattice deformation
Type of Principal strains Volume change
diffusionless
transformation type* order ** Sign Value
Shuffle C or D F All zero Zero Zero up to
S1 0
–5
Dilatational D F All positive or Large Large:
All negative 10
–1
Quasi-martensitic C or D F Mixed sign Small Small:
S10
–4
–10
–3
Martensitic D F Mixed sign or Large Small or large:
Zero and +, – 10
–2
–10
–1
* C: continuous, D: discontinuous
** F: first order, S: second orderwww.iran-mavad.com
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648. As regards SrTiO
3, the rotation-vibra-
tional modes of the oxygen atoms are fro-
zen into the low-temperature positions be-
low 110 K. The temperature dependence of
the softening, expressed by the square of
the eigenfrequency of the mode, is repre-
sented schematically in Fig. 9-8 for a sec-
ond- and first-order phase transformation.
Usually, the low-temperature phase also
shows a soft mode as the temperature is
raised towards T
c.
Lattice softening can also be treated in
terms of a static approach in which the
stability of the lattice is examined when
submitted to small static or quasi-static ho-
mogeneous strains. The free energy is then
expressed as a function of the elastic con-
stants; for a lattice to be stable when sub-
mitted to small homogeneous strains, the
free elastic energy must increase for all
possible strains. For a cubic crystal this is
mathematically expressed by saying that
all the eigenvalues of the elasticity tensor
must be positive, in other words C
44>0,
(C
11–C
12) > 0, and (C
11+2C
12)>0.
The tendency toward mechanical in-
stability can also be studied through exam-
ination of the phonon dispersion curves,
which gives the relationship between the
wavevector qof the vibrational mode and
its eigenfrequency. The lattice instability
can correspond to uniform (q= 0) or mod-
ulated (q= non-zero) atom displacements
and the soft phonon may belong to an optic
or an acoustic branch; an example of a
measured dispersion curve is given in Fig.
9-9. The longitudinal acoustic branch in
zirconium shows a dip at about 2/3 [111],
which is the mode needed to transform the
high-temperature b.c.c. structure of Zr into
the omega structure. The slope of the trans-
verse acoustic branch of Nb
3Sn is very flat
at the origin on approaching the transition
temperature of 46 K (Shapiro, 1981). This
corresponds fairly well with the experi-
mental observation of a vanishing value of
the elastic shear constant C¢=(C
11–C
12)/2.
The atom displacements induced by this
soft mode coincide exactly with those as-
sociated with the deformation from cubic
598 9 Diffusionless Transformations
Figure 9-8.Temperature
dependence of the squared
frequency of the softening
mode for (a) a second-order
and (b) a first-order transfor-
mation, T
cand T
tbeing the
critical and the transforma-
tion temperatures, respec-
tively. (c) Phonon energy of
SrTiO
3measured below and
above T
c(after Rao and Rao,
1978).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

649. 9.3 General Aspects of the Transformation 599
to tetragonal structure. For Cu–Zn–Al no
soft mode is present at 2/3 [111], although
the LA branch shows an anomalous dip;
the branch measured perpendicular to it
proves that the point in the reciprocal space
is a saddle point and not a minimum. The
branch TA2 [110] (polarization [11
–
0]),
however, shows a small slope correspond-
ing to a low value of C ¢(Guénin 1982).
Transformation models have been pro-
posed for Cu–Zn–Al taking into account
both the anomalous dip and the small
slope.
In a number of materials undergoing
a cubic to tetragonal transformation, a
tweed-like pattern is observed in the parent
phase by transmission electron micros-
copy. This tweed contrast is characterized
by a ·100 Òdirection of the modulation, a
type of {110} ·11
–
0]Òshear strain and a
modulation which is incommensurate with
the parent phase. It is still debated whether
all the pre-transformational or precursor
effects are evidence of stable or metastable
modulated phases or whether they are well-
defined artefacts determined by the kinet-
ics of nucleation and the growth process.
A long-standing issue with bCu-, Ag-
and Au-base alloys that has now been re-
solved is the appearance of extra maxima
in the electron diffraction patterns of the
parent phase from quenched alloys. Over
the years these maxima have been given
various interpretations, often with an over-
emphasis as possible pre-martensitic ef-
fects. Systematic investigation, however,
established that these effects are in fact ob-
tained in martensitic structures located on
the surface of the thin foil and extending
inwards to a depth of 1 µm.
When considering martensitic transfor-
mations, the role played by the combina-
tion of lattice defects and of lattice instabil-
ities is of particular interest; the large defor-
mations present around the defects may in-
duce a localized lattice instability (Guénin
and Gobin, 1982), which may trigger the
nucleation of martensite on further cooling
or stressing.
Pre-transformational lattice instabilities
and soft modes and their relation to diffu-
sionless phase transformations have been
reviewed by Delaey et al. (1979), Nakani-
shi (1979), Vallade (1982), Nakanishi et al.
(1982), and Barsch and Krumhansl (1988).
The validity of the soft phonon or soft elas-
tic stiffness approach to martensite is a dif-
ficult and somewhat controversial subject.
9.3.3 Transformation Mechanisms
It should be emphasized that the pure lat-
tice distortions considered above do not
necessarily imply the actual path the atoms
follow during the transformation. For sec-
ond-order phase transformations, there is a
continuous change throughout the crystal
with decreasing temperature starting at T
c.
Following the soft-mode concept, the ap-
pearance of the new phase is considered as
the freezing of a particular wavelength vi-
bration. The Bain-type strain for a second-
order cubic to tetragonal transformation,
for example, is equivalent to two {110}
·11
–
0Òshear strains whose corresponding C¢
Figure 9-9.Phonon dispersion curves for b.c.c. Zr.
A pronounced dip occurs in the longitudinal (l) pho-
non dispersion curve in the vicinity of q= 2/3 [111]
(Sikka et al., 1982).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

650. the fully formed product. The exact mecha-
nisms for the various types of martensitic
transformations are still under debate.
9.3.4 Microstructures
The microstructures that result from
diffusionsless phase transformations show
typical features, which are related to the
crystallograph of the transformation. The
transformation is associated with a reduc-
tion in symmetry; consequently, different
equivalent orientational states of the prod-
uct phase are formed. A single crystal of
the parent phase thus transforms to a col-
lection of the product-phase crystals, called
variants, that are separated by interfaces.
The higher the symmetry of the parent
phase and the lower the symmetry of the
product phase, the greater is the number of
equivalent paths of transformation. The
number of equivalent orientations or vari-
ants is also determined by the symmetry
elements that are maintained or broken due
to the Bain strain. The collection of vari-
ants constitutes the microstructure.
The order of the transformation (whether
first or second order) also determines the
microstructure: in the former case parent/
product or heterophase interfaces in addi-
tion to product/product or homophase inter-
faces are created, whereas in the latter only
product/product interfaces are formed. In
the former case the first plates formed can
grow to a larger extent than those formed
later, which can lead, for example, to mi-
crostructures with fractal characteristics.
Fig. 9-10 shows a selection of character-
istic microstructures obtained through dif-
fusionless phase transformations.
9.3.5 Shape Changes
If we could transform a single crystal of
the parent phase into a single crystal of the
600 9 Diffusionless Transformations
shear constant vanishes at T
c. The trans-
formation mechanism is therefore not a
combination of expanding and contracting
atom movements, but a lock-in of long-
wavelength shear-type movements on
{110} planes in the ·11
–
0Òdirections in this
scheme.
As for the martensitic transformations,
the situation is not so straightforward. The
Bain-type strains are concerned only with
the correspondence between initial and fi-
nal lattices; they do not give the actual ob-
served crystal orientation relationships
between them. Based on the experimen-
tally determined orientation relationships,
different transformation mechanisms have
been proposed, such as shears on the planes
and along the directions involved in the
orientation relationship. However, these
shear mechanisms have been found to be
too simple to be consistent with the experi-
mental facts. More recently, a transforma-
tion mechanism has been proposed for
martensitic transformations of b.c.c. to
close-packed structures; a condensing state
of some soft phonon modes combined with
a homogeneous shear explains the variety
of structures that are found. For the same
transformations, Ahlers (1974) proposed a
two-shear mechanism; the first shear
creates the close-packed planes, whereas
the close-packed structure is obtained by
the second shear.
Martensitic transformations are first-
order phase transformations that occur by
nucleation and growth. The growth stage
generally takes place by the motion of
interfaces converting the parent phase to
the fully formed product phase. Two types
of paths have to be considered for the case
of nucleation, the “classical” and the “non-
classical” nucleation paths (Olson and Co-
hen, 1982). The latter involves a continu-
ous change in structure whereas the former
involves a nucleus of the same structure aswww.iran-mavad.com
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651. 9.3 General Aspects of the Transformation 601
Figure 9-10.A selection of
microstructures obtained by
diffusionless phase transfor-
mations: (i) schematic repre-
sentation of the microstruc-
ture of (a) martensite and
austenite, (b) the Dauphiné
twins between the low tem-
perature a- and the high
temperature b-phase of
quartz, and (c) the zig-zag
domain structure in neody-
mium pentaphosphate which
undergoes an orthorhombic
to monoclinic transforma-
tion (after James, 1988);
(ii) optical and transmission
electron micrographs of (a, b)
the twinned orthorhombic
YBa
2Cu
3O
xhigh-tempera-
ture superconductor (cour-
tesy H. Warlimont, 1989),
(c) the domain in SiO
2at the
transition temperature ato b
(846 K) (courtesy Van Ten-
deloo, 1989), and (d) the
fractal nature of the marten-
site microstructure in steel
(courtesy Hornbogen, 1989).www.iran-mavad.com
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652. product phase, the macroscopically visible
shape change would clearly reflect the
Bain-type strain; it is then the maximum
transformation-induced shape change that
can be achieved. Depending on the symme-
try relationship, this deformation can be
obtained in as many orientations as product
variants exist.
In a martensitic transformation, the mac-
roscopic shape change associated with the
formation of a single martensite plate is not
only the result of the Bain strain but also of
a lattice-invariant deformation. The total
macroscopic shape change is mainly a
shear deformation along the habit plane of
the martensite variant. The martensite plate
contains either a large number of stacking
faults or has twins inside. It is therefore not
a single crystal. If the single martensite
plate has twins inside and is subjected after
the transformation to an externally applied
stress, an additional shape change is ob-
tained by detwinning. Only then is the final
product a single crystal of the product
phase. Fig. 9-11a shows the changes in
shape after transforming a b-Cu–Zn–Al
single crystal into a single martensite vari-
ant and Fig. 9-11b shows the shape change
after partially transforming an iron whisker.
The transformed sample usually contains
a very large number of single-product do-
mains arranged in a special configuration.
In some systems the domains are arranged
such that the shape changes are mutually
accommodated. Because each product
variant is associated with a differently
oriented shape change, applying a mechan-
ical stress during the transformation will
promote the formation of those variants
that accommodate the applied stress. This
provides a resolved shape change in the di-
rection of the applied stress. This is the
fundamental concept of the shape-memory
effect, as will be explained further in Sec.
9.10.2.
602 9 Diffusionless Transformations
Figure 9-11.Macroscopic shape change associated
with martensite: (a) a Cu–Zn–Al single crystal before
and after transforming to martensite and (b) a par-
tially transformed Fe whisker (courtesy Wayman,
1989).
a
bwww.iran-mavad.com
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653. 9.3 General Aspects of the Transformation 603
If a single crystal of the product phase is
mechanically strained it can either be trans-
formed to another or be deformed to a dif-
ferently oriented single crystal of the prod-
uct phase (Fig. 9-12). Similar behaviour is
also typical of a number of ferroelastic
materials; the reorientation is there referred
to as “switching” (Wadhawan, 1982). The
switching force in these materials is not
only a mechanical stress but can also be an
electric or magnetic field, the domains be-
ing either electrically or magnetically po-
larized.
In first-order phase transformations, as
shown above, the full transformation shape
change is induced locally and is gradually
spread over the whole sample within a
small temperature interval, whereas in a
second-order phase transformation the
sample changes its shape homogeneously
and continuously as soon as the critical
transition temperature T
cis reached.
Until now, shape changes have been dis-
cussed that are induced by the forward
transformation. It is evident that if a sam-
ple of the low-temperature phase, a single
crystal or a polyvariant, is heated to tem-
peratures above the reverse transformation
temperature, similar shape changes are ob-
served, provided that the reverse transfor-
mation is also diffusionless. The situation
for second-order phase transformations is
straightforward; the sample whose shape is
changed during the forward transformation
and possibly after deformation below T
cre-
verts back to its original shape above T
cin
a homogeneous and continuous way. For
first-order transformations, the reverse
transformation is more complex and not
yet well understood. Much depends on
whether the forward transformation is
completed or not, and whether the growth
of the martensite plate occurs by bursts or
under thermoelastic conditions (see be-
low). Occurrence of the reverse transfor-
mation does not necessarily imply that the
original shape is restored. Depending on
the crystal symmetry of the product phase,
more than one crystallographically equiva-
lent path can be followed for the reverse
transformation. The shape changes that oc-
cur during the reverse transformation are at
the origin of the shape-memory effect and
are discussed in Sec. 9.10.2.
Figure 9-12.A series of macrographs representing the shape change while mechanically straining a Cu–Al–Ni
martensite single crystal; (a) to (e) increasing with time (Ichinose et al., 1985).www.iran-mavad.com
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654. 9.3.6 Transformation Thermodynamics
and Kinetics
A diffusionsless phase transformation
may be of second or first order. The former
is generally dealt with in the phenomeno-
logical Landau theory, while the latter
is treated by classical thermodynamics
(Kaufman and Cohen, 1958). The Landau
theory has, however, been extended to also
cover first-order phase transformations
(the Devonshire–Ginzburg–Landau theory)
and has been applied by Falk (1982) to
martensitic transformations. The reader is
referred to the chapter by Binder (2001) for
an introduction to those theories.
The chemical driving force occupies a
key position in the classical thermodynam-
ics of first-order diffusionless phase trans-
formations, a subject that is introduced in the
first chapter of this volume (Pelton, 2001). In
the following section, therefore, only those
aspects directly relevant to diffusionless
phase transformations will be dealt with.
Because no chemical composition
change is associated with a diffusionless
phase transformation, the parent and prod-
uct phases have the same homogeneous
chemical composition and hence they are
treated as a single-component system. For
those phase transformations whose structu-
ral change is easily described by a dis-
placement parameter, a phenomenological
description of the free energy as a function
of the order parameter in terms of the Lan-
dau theory leads to some interesting con-
clusions. In the following, the free energy
is discussed as a function of temperature
and composition. Other possible intensive
thermodynamic state variables include ex-
ternal pressure, mechanical stress, and
magnetic and electrical field strength.
The changes in chemical Gibbs energy,
DG, as a function of temperature and com-
position are shown schematically in Fig.
9-13 for first-order diffusionless phase
transformations between a parent phase,
denoted P, and a product phase, denoted M.
The product phase may be one of the low-
temperature equilibrium phases or a meta-
stable phase.
Taking again the martensitic transforma-
tion as an example, the transformation
starts at M
s, which is lower than T
0, and
finishes at M
f. This means that a higher
driving force is needed to complete the
transformation. On heating a fully marten-
sitic stress-free single crystal, the reverse
transformation sets in at a temperature A
s,
604 9 Diffusionless Transformations
Figure 9-13.Schematic representation of the molar
Gibbs energy (a) as a function of temperature but
constant composition and (b) as a function of compo-
sition for an Fe–Ni alloy with T
4>T
3>T
2>T
1and
T
2=T
0for X
Ni= X (after Mukherjee, 1982).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

655. 9.3 General Aspects of the Transformation 605
which is higher than T
0. The difference
between the forward and the reverse trans-
formation temperatures is the transforma-
tion hysteresis. The true first-order equilib-
rium temperature, T
0, which is calculated
from DG= 0, can thus only be bracketed
from experimental data for the forward and
the reverse transformation temperatures,
and is not necessarily halfway between M
s
and A
s.
Strain energy resulting from the transfor-
mational shape change and interfacial en-
ergy have been omitted from the free-en-
ergy curves in Fig. 9-13. These two non-
chemical-energy terms have to be consid-
ered, however, in the overall free-energy
balance. The strain energy associated with
the formation of a single domain of the
product phase is proportional to the volume
of that domain. The interfacial energy is
not directly related to the volume of the
transformed domain but merely to its sur-
face-to-volume ratio, and, in the case of an
anisotropic interfacial energy, also to the
orientation of the interface. Both quantities
are positive and thus consume part of the
chemical driving force for a forward trans-
formation. Both terms will, however, in-
crease the driving force for the reverse
transformation, provided that the inter-
facial coherence is not lost. The reverse
transformation might start below T
0if a
negligible net driving force is required for
nucleation.
Considering the Gibbs energy per unit
molar volume, the total Gibbs energy
change per unit molar volume for the for-
mation of a single domain of the product
phase embedded in the matrix phase is
given by
DG
tot=DG
chem+(DG
elast+jDG
surf) (9-3)
where
jis the surface-to-volume ratio
of the single domain. The two terms in
parentheses are then the non-chemical con-
tributions to the Gibbs energy change,
DG
non-chem, and Eq. (9-3) then becomes
DG
tot=DG
chem+DG
non-chem (9-4)
The transformation then proceeds until
DG
totbecomes minimum or, if the phase
boundary is mobile, until the total force at
the parent-to-product interface is zero. If
the advancing direction of the interface is
x, we can write
[∂(DG
tot)/∂x]dx= 0 (9-5)
or
[∂DG
chem/∂x]dx+[∂DG
non-chem/∂x]dx=0
The sum of the non-chemical restoring
forces is then identical with the chemical
driving forces. The difficult task now is
to find expressions representing the non-
chemical terms. Three thermodynamic ap-
proaches have been worked out, dealing es-
sentially with the influence of the two non-
chemical contributions on the transforma-
tion behavior (Roitburd, 1988; Ball and
James, 1988; Shibata and Ono, 1975, 1977).
According to Roitburd (1988), the strain
energy, which arises in crystals owing to a
diffusionless phase transformation, can de-
crease if the crystals are subdivided into
domains arranged such that a maximum
compensation of the individual strain fields
is achieved. In order to determine which ar-
rangements are energetically most favor-
able, Roitburd calculates the strain energy
for arbitrary domain arrangements, and
then minimizes this energy. The formula-
tion of this problem is complex and can
hardly be solved in general, but he suc-
ceeded for some specific cases.
Ball and James (1988) do not assume
any geometric restrictions on the shape or
arrangements of the domains; they found
this necessary to determine microstructures
occurring in complex stress fields, or to
explore new and unusual domain arrange-www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

656. ments. The general aim of their work was
to develop mathematical models, using cal-
culus of variations, capable of predicting
the microstructure, especially the micro-
structural details at the interface between
the parent and the product phases. At-
tempts have been made to predict the pos-
sible interfaces between austenite and mar-
tensite from a minimization of a Gibbs en-
ergy function, which depends on the defor-
mation gradients of all possible domain
variants and on temperature. A deforma-
tion or domain is then termed stable if
it minimizes the total energy. They show,
among others, that a martensite–austenite
interface can exist as an energy-minimiz-
ing sequence of very fine twins. A further
example of an intriguing application is
the formation of triangular Dauphiné twins
in quartz, which become finer and finer in
the direction of increasing temperature. A
Gibbs energy function accounting for this
behaviour could be constructed.
Shibata and One (1975, 1977) use the
Eshelby theory; the principle of their calcu-
lation is in a corrected version (Christian,
1976) illustrated schematically in Fig.
9-14. An embedded part of the parent
phase is cut out (step a) and is allowed to
transform stress-free into the product phase
(step b). A lattice-invariant deformation is
applied (step c) and the transformed crystal
is subjected to forces along its surface such
that it is deformed to the original shape
(step d). The thus deformed part of the
product phase is introduced in the empty
space of the parent phase (step e), and the
forces are removed, creating internal
stresses in both the product and the parent
phase. The total energy is then calculated
as a function of all possible lattice orienta-
tions, taking into account the actual elastic
constants and the modes of lattice-invari-
ant deformation, twinning or slip.
The total Gibbs energy of the system is
therefore a function not only of the intrin-
sic energies of the stress- and defect-free
parent and product phases, but also of the
arrangement of the domains. The non-
chemical component of the total Gibbs en-
ergy of the transforming system is lowered
by an appropriate rearrangement of the mi-
crostructure and/or by irreversible plastic
deformation.
If the structural change can be repre-
sented by an order parameter e, the Gibbs
energy of the system can then, according to
the theory of Landau–Devonshire, be rep-
resented by
G=G
0+a(T–T
1)e
2
–Be
4
+Ce
6
(9-6)
where a, Band C are constants and T
1>0.
It can be shown that the high-temperature
phase becomes unstable with respect to
any fluctuation of e, as soon as the temper-
ature reaches T
1on cooling, and hence is
thermodynamically metastable between T
0
and T
1. Accordingly, the low-temperature
606 9 Diffusionless Transformations
Figure 9-14.The necessary steps in calculating the
elastic stresses induced by a transforming ellipsoid
(Christian, 1976). (For details see text.)www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

657. 9.4 Shuffle Transformations 607
phase cannot exist at temperatures higher
than T
2, which is the temperature above
which the low-temperature phase becomes
unstable with respect to any fluctuation in e.
That additional undercooling is needed
for further transformation below M
sis due
(in part) to the non-chemical contributions,
which increase with increasing volume
fraction of transformed product.
An interesting aspect of the diffusionless
phase transformations that are accompa-
nied by a volume and shape change is the
role played by external stresses, e.g., hy-
drostatic or uniaxial. Both thermodynamics
and experiments show that the transforma-
tion temperatures are affected by the appli-
cation of stresses. According to Wollants et
al. (1979), the relationship between a uni-
axially applied stress
sand the transfor-
mation temperature Tdepends on the
transformation entropy and the transfor-
mational strain in the direction of the ap-
plied stress. This relationship, the Clau-
sius–Clapeyron equation for uniaxially
stressed diffusionless first-order phase
transformations, is
d
s/dT=–DS/ e=–DH*/[T
0(s)e] (9-7)
where DH*=DH–FDl=DH–
seV
m=
T
0(s)DSis itself a function of the applied
load,
e=Dl/l, lis the total “molar length”
of the sample, and Fis the applied load
(
s=F/A). This equation is similar to that
relating the equilibrium temperature to the
hydrostatic pressure, except for the nega-
tive sign on the right-hand side of Eq. (9-7).
This relationship between d
sand dTis ex-
perimentally constant for most of the diffu-
sionless transformations, which means that
the thermodynamic quantity DSis, within
the experimental scatter, independent of
temperature and stress. Knowing the trans-
formation strain, uniaxial tensile tests are
very useful for determining the transforma-
tion entropy.
The most relevant thermodynamic data
for the various diffusionless phase transfor-
mations are presented in Table 9-2.
9.4 Shuffle Transformations
Shuffle transformations from a distinct
class of diffusionless phase transitions. At
the unit-cell level the atom displacements
are intercellular with little or no pure strain
of the lattice. The role of elastic strain en-
ergy in shuffle-phase transformations is
sufficiently small that the transformation
can either occur continuously from the par-
ent to the product phase or that it is com-
pletely controlled by interfacial energy. In
the former case the transformation is sec-
ond order whereas in the latter it is a first-
order phase transformation.
Cohen et al. (1979) gave three examples
which clearly illustrate the shuffle transfor-
mations. The displacive transformation in
strontium titanate is the prototype example
of a pure shuffle transformation. The asso-
ciated strain energy is so small that the
transformation occurs continuously. The
b-to-w transformation in some Ti and Zr
alloys shows, in addition to the shuffle
displacements, small homogeneous lattice
distortions. These distortions are small
enough for the transformation mechanism
and the resulting microstructure to be dom-
inated only by shuffling. In ferroelectric
transformations, which are accomplished
by shuffling, the interfacial energy is con-
stituted largely by electrostatic interaction
energies and is therefore dependent on the
orientation of the domain interfaces. The
interfacial energy in those materials is
strongly anisotropic and controls the poly-
domain structure.
Phase transformations that can be en-
tirely described by shuffle diplacements
are often found where the change in crystalwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

658. structure is such that the point group to
which the crystal structure of the product
phase belongs is a subgroup of that of the
parent phase. In other words, some symme-
try elements of the high-temperature phase
are lost on cooling below the transition
temperature T
c. Because of this group/sub-
group relationship, the product phase pos-
sesses two or more equally stable orienta-
tional states in the absence of any external
field. The change in crystal structure can
easily be described by an order parameter
which itself is related to the shuffle dis-
placement. For strontium titanate, the order
parameter would then simply be the rota-
tion angle that describes the displacement
of the oxygen atoms around the titanium
atoms (see Fig. 9-4). For convenience, the
order parameter is taken as zero for the
high-temperature configuration and as non-
zero for the low-temperature phase. The
majority of such transitions are found in
chemical compounds (e.g., Rao and Rao,
1978). As soon as the critical temperature
T
cis reached on cooling, the order parame-
ter changes continuously. The thermody-
608 9 Diffusionless Transformations
Table 9-2.The elastic shear constant and some thermodynamic data characterizing the diffusionless phase
transformation (Delaey et al., 1982b).
Elastic shear constant C ¢ Trans- Thermodynamic quantities
near the M
stemperature formation
strain Heat of Change Chemical Transfor-
C¢(1/C¢) (dC¢/dT) transforma- in driving mation
(10
10
Pa) (10
–4
K
–1
) tion entropy force tempera-
(J/mol) (J/(K · mol)) (J/mol) ture hys-
teresis (K)
Ferrous g
Æa¢2–3 negative ≈10
–1
≈2000–300 5.8 150–450 200–400
g
Æe 3–10 (positive 600–1800
for Ni > 30%)
Co alloys negative ≈10
–3
≈400–500 ≈0.2 4–16 40–80
rare-earth
alloys
Ti and Zr 0.1 negative ≈2¥10
–2
≈4000 ≈1.0 ≈25 –
alloys bCu–Ag–Au 0.5–1 4–20 ≈10
–2
≈160–800 0.2–3.0 ≈8–20 10–50
alloys (positive)
In alloys 0.05–01 ≈1000 ≈10
–3
≈0– ≈1.5 1–10
(positive)
Mn alloys positive ≈10
–3
(strongly)
A 15 com- 0.5 ≈1000–3000 ≈10
–4
pounds (positive)
Fe–Pt ≈320 – ≈16 ≈20–200
(ordered) (ordered)
Fe–Pd ≈1 ≈100 ≈10
–3
≈1200 ≈1200
alloys (positive) –10
–2
(disordered) (disordered)www.iran-mavad.com
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659. 9.4 Shuffle Transformations 609
namics of such transformations are then in
the temperature range close to T
c, which is
dealt with by a Landau approach.
9.4.1 Ferroic Transformations
Usually a phase transition that is domi-
nated by shuffling is associated with a
change in some physical properties, such as
spontaneous electrical polarization, strain
and magnetization. Because the crystal
symmetry of the parent phase decreases
during the phase transition, two or more
equivalent configurations of the product
phase are formed. In the absence of any ex-
ternal field, the average polarization of the
product phase is zero. However, under a
suitably chosen driving force, which may
be an electrical field (E), a mechanical
stress (
s), or a magnetic field (H), the do-
main walls of the product phase move,
switching the crystal from one domain or-
ientation to the other. Owing to the applica-
tion of a uniaxial stress, for example, one
orientation state can be trensformed repro-
ducibly into the other, and the crystal is
then said to be “ferroelastic”. The materials
exhibiting this property are called ferro-
elastic materials. Similarly, we can define
ferroelectric and ferromagnetic materials.
According to Wadhawan (1982), “phase
transitions accompanied by a change of the
point-group symmetry are called ferroic
phase transitions. We refer to a crystal as
being in a ferroic phase if that phase results
from a symmetry-lowering ferroic phase
transition”.
Not all ferroelastic phase transitions be-
long to shuffle transformations as defined
in Fig. 9-1. Indeed, in addition to shuffle
displacements, as for example those in-
volved in the cubic to tetragonal transition
in barium titanate, the lattice may become
homogeneously distorted. For the example
considered here, the lattice distortion oc-
curs discontinuously at the transition tem-
perature; the lattice parameters change
abruptly (Fig. 9-15). Even below this tran-
sition temperature, the lattice continues to
be homogeneously distorted. In cases
where this lattice is tetragonal, the c/aratio
steadily increases. For barium titanate the
change in c/acontinues until the tempera-
ture for another first-order phase transition
is reached. Many such phase transforma-
tions are encountered in chemical com-
pounds. In some cases the amount of spon-
taneous strain is not large enough to con-
trol the microstructure. In others, the strain
energy associated with the transformation
will be dominant. The transformation is
then, according to Fig. 9-1, quasi-marten-
sitic or martensitic. In ferroelectric materi-
als, the interfacial energy also has to be
Figure 9-15.Temperature de-
pendences of the lattice
parameters of the different
phases of BaTiO
3.www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

660. taken into account and may even become
the dominant parameter controlling the
microstructure. A ferroic ferroelastic phase
transformation can thus be a shuffle, a
quasi-martensitic, or a martensitic phase
transformation, a clear discrimination is
only possible by analyzing all the transfor-
mation characteristics, and this not after
but during the transformation.
9.4.2 Omega Transformations
The omega transformation is known to
occur as a metastable hexagonal or trigonal
phase in certain Ti, Zr and Hf alloys on
cooling from the high-temperature b.c.c b-
phase solid solution or as a stable phase
under the influence of high hydrostatic
pressures or shock waves. The w-phase
cannot be suppressed by quenching and
forms as small cuboidal or ellipsoidal par-
ticles with a diameter of 10–20 nm. Its lat-
tice is obtained by collapse of one pair of
(111) planes of the parent b.c.c. b-phase,
leaving the two adjacent planes unaltered
(Fig. 9-4b). The collapse can be repre-
sented as a short-wavelength displacement
of atoms. The displacement of the atoms
occurs over a distance approximately equal
to 2/3 ·111Ò. Each lattice site can thus be
associated with a forward, zero or back-
ward displacement that can be represented
by a sinusoidal wave dividing the repeat
distance along a [111] direction into six
parts. The collapse is not always complete,
and then results in a “rumpled” plane. If the
collapse is incomplete the crystal structure
of the w-phase is trigonal; if the collapse is
complete, it is hexagonal. The figure also
shows that reversing the direction of the dis-
placement will not lead to a collapse of the
{111} planes. Moreover, it can be shown
that a 2/3 [111] displacement wave is equiv-
alent to a 1/3 [1
–
12] displacement wave.
If the displacement is taken as the order
parameter in a Landau-type approach, the
transformation is seen to be first order and
the Gibbs energy as a function of this order
parameter has an asymmetric shape. Con-
sequently, no negative values of the order
parameter are then allowed (Fig. 9-16).
The b-to-wtransformation can also be par-
aphrased in terms of a soft mode. The lat-
tice tends to a mechanical instability for a
2/3 ·111Òlongitudinal mode. This tendency
can be shown when measuring the phonon
dispersion curves by inelastic neutron scat-
tering. Such curves are reproduced in Fig.
9-9a for zirconium; a clear dip is visible at
the 2/3 [111] position.
The omega transformation has been re-
viewed by Sikka et al. (1982).
9.5 Dilatation-Dominant
Transformations
A transformation is regarded by Cohen
et al. (1979) as dilatation dominant if no
610 9 Diffusionless Transformations
Figure 9-16.Gibbs energy change for the b.c.c. to w
transformation as a function of the order parameter
for various reduced temperatures (after de Fontaine,
1973).www.iran-mavad.com
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661. 9.6 Quasi-Martensitic Transformations 611
undistorted line can be found in the lattice-
distortive deformation. The f.c.c.-to-f.c.c.¢
transformation in cerium is considered as
the prototype for a dilatational dominant
transformation. Below 100 K cerium un-
dergoes a pure volume contraction of about
16%; the ellipsoid of the f.c.c.¢phase thus
falls completely inside that of the high-
temperature f.c.c. phase. The low-tempera-
ture cubic to tetragonal transformation in
tin also appears to be dominated by dilata-
tion, although some deviatoric components
are present; the volume expansion is about
27%. The deviatoric component is not
large enough to let the original sphere
intersect with the dilated ellipsoid.
The name “dilatational diffusionless
phase transformation” has been used by
Buerger (1951) but with a different mean-
ing. In the systems he considers, for exam-
ple the CsCl-to-NaCl transitions in many
alkali metal halides, he defines the term
dilatational as follows: “the transformation
can be achieved by a differential dilatation
in which the structure expands along the
trigonal axis and contracts at right-angles
to the axis”. Although the volume change
in these and other related inorganic sys-
tems may be very large (up to 17%), the
transformation is, in the context of Fig. 9-1,
clearly not dilatation dominant but devi-
atoric dominant. See Kriven (1982) for a
more detailed review of these dilatational
dominant transformations.
9.6 Quasi-Martensitic
Transformations
The quasi-martensitic and the marten-
sitic transformation are both deviatoric
dominant and are characterized by an un-
distorted line. The morphologies of the
product phases of the two transformations
are very similar (large plates, occurrence of
variants and twins). A distinction between
the two transformations cannot be made by
simply judging only the product morphol-
ogy, but rather a knowledge is required of
the morphological relationships between
parent and product phases during the trans-
formation itself. It may be adequate to say
first what a quasi-martensitic transforma-
tion is: a quasi-martensitic transformation
is not a martensitic transformation, which
itself is “a first-order phase transformation,
that undergoes nucleation, passes through a
two-phase mixture of the parent and prod-
uct phases, and which product grows with a
transformation front in a plate-like or lath-
like shape being indicative of a tendency
toward an invariant-plane interface” (Co-
hen et al., 1979). If a deviatoric dominant
transformation does not satisfy the above
criterion, it should not be designated as
martensitic but as quasi-martensitic.
Three aspects are common to most of the
materials that transform quasi-martensiti-
cally: (1) the lattice distortion is small and
deviatoric dominant and the change in lat-
tice distortion is continuous or nearly con-
tinuous; (2) a banded internally twinned
microstructure gradually builds up on cool-
ing below T
c; and (3) a mechanical lattice
softening is expressed by elastic shear
constants approaching zero as T
cis ap-
proached. Because of the small lattice dis-
tortion at the transformation, the ratio of
the strain energy to the driving energy for
transformation is small; this ratio has been
used by Cohen et al. (1979) as an alterna-
tive index to differentiate quasi-martensitic
transformations from martensitic.
The three aspects are now illustrated by
taking the manganese-based magnetostric-
tive antiferromagnetic alloys as an example
(see Delaey et al., 1982a). One of the four
polymorphic states of manganese is the
gamma f.c.c. phase which is stable only atwww.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

662. high temperatures. Alloying with elements
such as Cu, Ni, Fe, Ge, Pd and Au stabi-
lizes the f.c.c phase and the latter can be re-
tained by quenching. However, owing to
the antiferromagnetic ordering, the lattices
become homogeneously distorted. This or-
dering to the Mn atoms starts at a temper-
ature T
N, which is the Néel temperature
for the paramagnetic to antiferromagnetic
transition. The transformed product phase
has a banded microstructure containing
fine twins. The temperature at which this
banded microstructure is formed does not
always coincide with the transformation
temperature T
N. Vintaikin et al. (1979) di-
vide these antiferromagnetic alloys into
three classes according to the relative posi-
tions of the temperature T
Nand the temper-
ature T
TW. The latter is the temperature at
which the banded microstructure sets in.
Depending on the type of lattice distortion,
the alloys are grouped into three classes,
each class being characterized by the rela-
tive positions of the two temperatures. A
schematic representation of the phase dia-
gram of the Mn-based alloys is given in
Fig. 9-17a, showing the temperature–com-
position areas in which the various crystal
structures and microstructures are ob-
served. The accompanying variation in the
lattice parameters as a function of tempera-
ture for the three classes of Mn-based al-
loys is given in Fig. 9-17b.
The changes in lattice parameters show
that the transformation is almost second or-
der, except for some alloys of class I and III
where the transformation is weakly first
order. A phase transformation is called
“weakly first order” whenever the height of
the discontinuous jump in the correspond-
ing thermodynamic property is very small.
The formation of the twinned banded mi-
crostructure extends over the entire volume
of the sample quasi-instantaneously and is
visible in polarized light because of the
non-cubic structure of the product phase.
Similar microstructures are observed in
other quasi-martensitic product phases
such as V
54–xRu
46Os
x(Oota and Müller,
1987). The microstructure, if properly
oriented with respect to the prepolished
surface, exhibits a surface relief effect that
is enhanced as the temperature decreases
below T
TW. This surface relief proves that
the transformation is accompanied by a
shape change associated with each domain.
Because of the continuously changing lat-
tice parameters, accommodation stresses
are built up as the temperature decreases.
An appropriate arrangement of these do-
mains reduces the overall stored elastic en-
ergy; further changes in microstructure are
therefore expected to occur even below the
transition temperature.
Class II alloys do not exhibit the twinned
banded microstructure immediately below
T
N. In the temperature region between T
N
and T
TW, broadening of some of the X-ray
diffraction peaks is observed, which is
612 9 Diffusionless Transformations
Figure 9-17.Schematic representation of (a) the
phase diagram and (b) the variation of the lattice pa-
rameters for the three classes of Mn-based alloys
(Delaey et al., 1982a).www.iran-mavad.com
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663. 9.7 Shear Transformations 613
attributable to a chaotic distribution of the
aand caxes with small undercooling. At
T
TWthe banded structure becomes visible
(point A in Fig. 9-17) and a tetragonal
structure can now be clearly detected by X-
ray diffraction. If the sample is now heated,
only the banded microstructure disappears,
not at A but at a temperature B that coin-
cides with the Néel temperature. This
proves that on cooling, very small, submi-
croscopic tetragonal regions are first
formed as soon as T
Nis reached. Hocke and
Warlimont (1977) have shown that when
the distortion |c/a – 1|becomes greater
than 0.005, a critical value is obtained at
which the elastic strain is relaxed through
coalescence of the small distorted regions
into large banded twinned regions. Thus, at
T
TWthere is not a phase transformation but
a stress relaxation in the microstructure,
which results in a twinned microstructure.
The lattice distortive phase transformation
itself occurs at T
N, followed immediately
(class I) or after some undercooling (class
II) by a domain rearrangement and macro-
scopic twinning.
Similar conclusions can be drawn for
other quasi-martensitic transformations, as
for example in the iron–palladium alloys;
the Pd-rich f.c.c. phase transforms on cool-
ing first to an f.c.t phase and at lower tem-
peratures to a b.c.t. phase. The f.c.c.-to-
f.c.t. transformation, although sometimes
regarded as martensitic, shows all the char-
acteristics of a quasi-martensitic transfor-
mation.
Because the formation of each single do-
main is associated with a shape change and
thus with accommodation stresses, the ap-
plication of an external stress to the trans-
formed product will result in a macro-
scopic shape change. As the domain boun-
daries, which for the Mn-based alloys coin-
cide with the antiferromagnetic boundar-
ies, are mobile, the banded structure will
gradually disappear and the product phase
becomes a single domain maximizing the
shape change. The shape change thus ob-
tained is gradually recovered on heating
the sample and is completely recovered at
T
Nand not (as in the case of Mn-based
alloys of class II) at T
TW, but at the point B
in Fig. 9-17. The quasi-martensitic alloys
thus also exhibit the shape-memory effect.
Some of the materials characterized by
shuffle displacements during the phase
transition may develop elastic strains as
transformation proceeds. As in ferroelec-
trics, for example, in addition to the elastic
strain energy, the dipole interaction energy
also contributes to the polydomain forma-
tion. If the elastic strain energies are only
slightly dominating, the transformation is
quasi-martensitic; if, however, the elastic
strain energy is largely dominating, the
transformation can be martensitic.
Sometimes it becomes difficult to differ-
entiate between martensite and quasi-mar-
tensite, as for example in In-based alloys.
In particular, if quasi-martensitic samples
are cooled in such a way that a temperature
gradient is created across the sample, the
product phase and the parent phase then
coexist and apparently the transformation
goes through a two-phase region, the two
regions being separated by a blurred or
planar interface. Such observations do not,
of course, facilitate the distinction between
quasi-martensite and true martensite.
9.7 Shear Transformations
In this section we discuss a special group
of phase transformations, the so-called
shear transformations or polytypic transi-
tions, which strictly belong to the marten-
sitic transformations. According to Verma
and Krishna (1966), “polytypism may be
defined, in general, as the ability of a sub-www.iran-mavad.com
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664. stance to crystallize into a number of dif-
ferent modifications, in all of which two
dimensions of the unit cell are the same
while the third is a variable integral multi-
ple of a common unit. The different poly-
typic modifications can be regarded as
built-up of atom layers stacked parallel to
each other at constant intervals along the
variable dimension. The two unit-cell di-
mensions parallel to these layers are the
same for all the modifications. The third
dimension depends on the stacking se-
quence, but is always an integral multiple
of the layer spacing. Different manners of
stacking these layers may result in struc-
tures having not only different morpholo-
gies but even different lattice types and
space groups”. Some random disorder of
layers (faulted sequences) is almost always
present. Polytypic transitions are then tran-
sitions among different polytypes; the
movement of partial dislocations along the
basal plane constitutes the transition mech-
anism, thereafter the name shear trans-
formations. Polytypic transformations are
found in a variety of inorganic compounds
and also in metals and alloys.
Polytypic phases are constructed by
stacking basic units in a cubic, hexago-
nal or rhombohedral sequence. The stack-
ing sequences are described by three key
layer positions, X, Y and Z; a cubic se-
quence (C) is represented by the sequence
XYZXYZ …, a hexagonal (H) by, for
example, XYXZ … or XYXZXYXZ …,
and a rhombohedral (R) by, for example,
XYZYZXZXYXYZYZXZXY … . Many
other stacking variants are possible and
unit cells containing as many as 126 or 144
layers have been reported. Each unit itself
can contain a single layer, as in cobalt and
its alloys, or two as in silicon carbide.
Transitions between different modifica-
tions can be achieved either by a simple
shear, a shear combined with shuffle dis-
placements or the movement of partial dis-
loctions along the basal plane. For exam-
ple, the transition between a 2 H and a 3 C
stacking is easily performed by a shear,
whereby the basic units are kept together in
pairs (Fig. 9-18). This shear results in a
large deviatoric shape change. It should be
kept in mind that the interlayer spacing
need not be constant, as is observed in the
f.c.c.–h.c.p. changes in metals; the trans-
formation may involve small changes and
thus be IPS (invariant plane strain) rather
than simple shear transitions.
The same transitions can be achieved by
the generation and movement of closely
spaced and repeatedly arranged partial dis-
locations. The passage of a positive partial
dislocation shifts the crystal in the direc-
tion XÆYÆZÆX and negative partial
dislocations shifts the crystal in the direc-
tion XÆZÆYÆX. The following dis-
tribution of partial dislocations on the unit
layers in the direction perpendicular to the
layers is proposed by Liao and Allen
(1982) (a layer without a partial dislocation
is denoted by a dot):
· – + · for 2 HÆ4H
· – – – · · for 6 HÆ3C
The polytypic shear transformation from
one modification to the other is thus ac-
complished by a coordinated propagation
of groups of partials along the interface
between the two phases. The lateral dis-
614 9 Diffusionless Transformations
Figure 9-18.Mechanism of the f.c.c.-to-h.c.p. trans-
formation (Nishiyama, 1978).www.iran-mavad.com
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665. 9.8 Martensitic Transformations 615
placement of the interface, which is the
thickening of the new phase generated by
the movement of those partials, is then due
to the formation of partial dislocations and
their outward movement. During the tran-
sition from a cubic to a hexagonal 2H se-
quence, it has been implicitly assumed
above that the glide of the layers would al-
ways occur in the same direction. There ex-
ist, however, three different directions for
transforming an X stacked unit into a Z
stacked unit. If the glide occurs alternately
in these three directions, no shape change
results from such a mechanism, as shown
in Fig. 9-19 (Bidaux, 1988).
9.8 Martensitic Transformations
The characteristics necessary and suffi-
cient for defining a martensitic transforma-
tion are (a) displaciveness of the lattice-
distortive type involving a shear-dominant
shape change, (b) diffusion not required for
the transformation, and (c) sufficiently
high shear-strain energy in the process to
dominate the kinetics and morphology dur-
ing the transformation (Cohen, 1982). The
definition is thus not based on the identity
of the transformation product itself (its
structure, specific morphology or proper-
ties), but rather on how it forms.
The crystallographic and thermody-
namic aspects are fully discussed in the lit-
erature and have already been introduced
in a more general context in the above sec-
tions; only a brief overview is given here.
9.8.1 Crystallography of the Martensitic
Transformation
The relevant experimental observable
parameters of the martensitic transforma-
tion are the shape deformation, the habit
plane, the crystallographic orientation rela-
tionships, and the characteristic micro-
structures.
9.8.1.1 Shape Deformation
and Habit Plane
When a sample of the parent phase is
cooled to below M
s, a relief gradually ap-
pears on a prepolished surface of the par-
ent-phase crystal. The surface relief disap-
pears on heating to temperatures above A
s,
provided that no diffusion-controlled trans-
formation interferes.
The martensite phase usually takes the
form of plates; the plane of contact be-
tween the parent and the martensite phases
is called the “habit plane”. A schematic
representation of such a martensite plate
embedded in the matrix is shown in Fig.
9-20. During the formation of martensite,
straight lines (for example, scratches on the
prepolished surface) are transformed into
other straight lines and planes are trans-
formed into other planes. No discontinu-
ities are observed at the points of deflec-
tion. This distortion can thus be repre-
sented as a “linear homogeneous transfor-
mation” of vectors and can be expressed by
a matrix formulation. The macroscopic
shape deformation can be decomposed into
Figure 9-19.Two different mechanisms, (a) and (b),
to transform an h.c.p. to an f.c.c. structure (Bidaux,
1988).www.iran-mavad.com
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666. a component normal to the habit plane and
a shear component parallel to a shear direc-
tion located in this interface. The latter is
called “macroscopic shear” and quantifies
the shape deformation, whereas the former
represents the volume change associated
with the transformation. A careful analysis
of the surface relief reveals that the habit
plane itself is unrotated and that any vector
in this interface is also left unrotated and
undistorted by the shape change. The habit
plane is thus essentially “undistorted” and
the macroscopic shape change associated
with the formation of martensite is thus an
“invariant plane strain” deformation, ab-
breviated to IPS. The most general invari-
ant plane strain deformation, as observed
in most martensitic transformations, can be
achieved by combining an extension and a
simple shear.
The habit plane and the direction of mac-
roscopic shear are, with few exceptions,
not simple low-indexed crystallographic
planes or directions of the parent or prod-
uct phase. They are usually represented in
a stereographic projection as shown sche-
matically in Fig. 9-20.
9.8.1.2 Orientation Relationship
The next most important observable pa-
rameter is the crystallographically well-de-
fined “orientation relationship” that exists
between the lattices of the parent and the
martensite phases. It is described either by
the angles between certain crystallographic
directions in both phases or by specifying
the parallelism between certain planes and
directions. This parallelism does not need
to be rigorous, however, experimental re-
sults usually deviate slightly. Nevertheless,
the fact that such crystallographic parallel-
ism is observed yields important informa-
tion concerning the possible mechanisms
explaining the change in crystal structure.
Some of those relationships observed in
steels received great attention in the early
martensite literature. Depending on the
alloy composition, the f.c.c. austenite in
steels transforms either to a b.c.c. or b.c.t.
martensite or to an h.c.p. martensite, which
itself may further transform into b.c.c. mar-
tensite. As regards the transformation of
f.c.c. austenite to b.c.c. or b.c.t. martensite,
the orientation relationships are as follows:
– the Kurdjumov–Sachs (K–S) relations:
(111)
P//(011)
a¢and [01
–
1]
P//(11
–
1)
a¢
616 9 Diffusionless Transformations
Figure 9-20.Schematic representation of (a) a sin-
gle martensite plate embedded in a single crystal of
the matrix phase, (b) a twinned plate and the position
of habit and twin plane, and (c) their stereographic
representation.www.iran-mavad.com
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667. 9.8 Martensitic Transformations 617
– the Nishiyama–Wassermann (N–W) re-
lations:
(111)
P//(101)
a¢and [12
–
1]
P//(101
–
)
a¢
– the Greninger–Troiano (G–T) relations
(here the planes and directions are no
longer exactly parallel):
(111)
P≈(011)
a¢and [1
–
01]
P≈[1
–
1
–
1]
a¢
Table 9-3.The crystallographic observables of the martensitic transformations in some metals and alloys
(courtesy G. Guénin et al. 1979*).
Alloy system Structural change Composition wt.% Orientation relationship Habit plane
Fe–C f.c.c. 0–0.4% C (111)
PΩΩ(101)
M (111)
P
Ø [11¯0]
PΩΩ[11¯1]
M
b.c.tetr. K–S relationship
0.55–1.4% C K–S relationship (225)
P
1.4–1.8% C Idem
Fe–Ni f.c.c. 27–34% Ni (111)
PΩΩ[101]
M
Ø [12¯1]
PΩΩ[101¯]
M ≈(259)
P
b.c.c. N-relationship
Fe–C–Ni f.c.c. 0.8% C–22% Ni (111)
P≈1° of (101)
M (3, 10, 15)
P
Ø [12¯1]
P≈2° of [101¯]
M
b.c.tetr. G–T relationship
Fe–Mn f.c.c. 13 to 25% Mn (111)
PΩΩ(0001)
e (111)
P
Ø [11¯0]
PΩΩ[12¯10]
e
h.c.p. (e -phase)
Fe-Cr-Ni f.c.c. 18% Cr, 8% Ni (111)
PΩΩ(0001)
eΩΩ(101)
a¢ e(111)
P
Ø [11¯0]
PΩΩ[12¯10]
eΩΩ[11¯1]
a¢ a¢(111)
P
h.c.p. (e), b.c.c. (a¢)
Cu–Zn b b.c.c.
Æ9 R 40% Zn (011)
PΩΩ?(11
––
4)
M ≈(2, 11, 12)
P
Cu–Sn idem 25.6% SN [11¯1]
PΩΩ[1¯10]
M ≈(133)
P
Cu–Al b.c.c. 11.0 to 13.1% Al (101)
Pat 4° of (0001)
M 2° of (133)
P
Ø [111]
PΩΩ[101¯0]
M
h.c.p. distorted 12.9 to 14.7% Al (101 ¯)
PΩΩ(101¯1)
M 3° of (122)
P
[111]
PΩΩ[101¯0]
M
Pure Co f.c.c. (111)
PΩΩ(0001)
M (111)
P
Ø· 110Ò
PΩΩ[112¯0]
M
h.c.p.
Pure Zr b.c.c. (101)
PΩΩ(0001)
M (596)
P
Ø [111]
PΩΩ[112¯0]
M (8, 12, 9)
P
Pure Ti h.c.p. (334)
P
(441)
P
Pure Li Burgers relations
* Gobin, P. F., Guénin, G., Morin, M., Robin, M. (1979), in Transformations de Phases à l’État Solide-Trans-
formations Martensitiques. Lyon: Dep. Gènie Phys. Mat., INSAwww.iran-mavad.com
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668. Concerning the transformation of f.c.c. to
h.c.p. austenite and that of h.c.p. to b.c.c.
martensite (the b.c.c. to h.c.p. relation is
known as the Burgers relation), the follow-
ing relations apply:
(111)
P//(0001)
e//(101)
a¢
and
[11
–
0]
P//[12
–
10]
e//[111
–
]
a¢
Taking the N–W relations as an exam-
ple, any one of the four crystallographi-
cally equivalent {111} austenite planes,
(111), (1
–
11), (11
–
1) and (111
–
), can be the
plane of parallelism. In each such plane
any one of the three ·12
–
1Òdirections, which
happen to be directions of the Burgers vec-
tors, can be chosen. This therefore results
in 12 different orientations of an a¢-crystal
in one austenite crystal. These differently
oriented martensite crystals are called
“variants”. It can easily be shown that the
K–S relations lead to 24 variants.
Orientation relationships and the orien-
tation of the habit plane change from one
alloy system to another, and within a given
alloy system from one composition to an-
other. The observable crystallographic pa-
rameters are summarized in Table 9-3 for
a large number of alloy systems; a more
complete list of these and other crystallo-
graphic characteristics of various marten-
sites is given by Nishiyama (1978).
9.8.1.3 Morphology, Microstructure
and Substructure
Because the martensitic transformation
is a first-order phase transformation, both
phases, the parent and the martensite
phase, coexist on cooling in a temperature
range between M
sand M
fand on heating
between A
sand A
f. Martensite thus occurs
in physically isolated regions, the morphol-
ogy of which is typical of the transforma-
tion. This morphology is easily observed
by light optical microscopy (LOM) and the
mutual arrangement of these regions con-
stitutes the microstructure at the LOM
level. Electron microscopic analysis re-
veals that also at the submicroscopic level
martensite is characterized by a typical
substructure. The morphological, micro-
structural and substructural aspects of mar-
tensite are briefly discussed below.
The martensite regions are generally
plate-shaped, i.e. one lateral dimension is
much smaller than the other two. If the two
larger dimensions are nearly equal they are
called “plates”, and if they are very un-
equal “laths”. A typical lath in low-carbon
steel (with a carbon content less than 0.4%)
has dimensions 0.3¥4¥200 µm
3
. How-
ever, martensite formed into the parent
phase does not always appear as a geomet-
rically well-shaped plate. Martensite plates
that form near a free surface or in a single
crystal as a result of a single-interface
transformation may show the idealized
plate-like shape. In such a single-interface
transformation the habit plane extends
from one side of the crystal to the other
(see Fig. 9-11). Because of the shape
change and the high elastic stresses that are
created, a thick plate cannot terminate in-
side a parent crystal. As is frequently ob-
served, lenticular shapes or groupings of
differently oriented martensite plates will
reduce these elastic stresses.
In the case of lenticular martensite, the
habit plane is no longer a plane but a
curved surface and the normal average of
the lenticular plate is then taken as the or-
ientation of the habit plane. Sometimes this
orientation is visible as a “midrib” in some
martensites (Fig. 9-21). It is believed that
the martensite could grow to a certain ex-
tent as a plate, but that lenticular shapes are
formed owing to the high elastic stresses
that are building up. The high stresses may
618 9 Diffusionless Transformationswww.iran-mavad.com
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669. 9.8 Martensitic Transformations 619
trigger other plates to form in the vicinity
of the plate formed earlier, giving rise to an
“autocatalytic growth” of martensite.
A multivariant martensite arrangement
is the most commonly observed micro-
structure. Often variants are arranged in
some recognizable patterns and at times
numerous variants present in a regular ar-
ray give the impression of a martensite col-
ony. The latter is typical of the “massive”
microstructure, consisting of a packet of
parallel martensite “laths” separated by
more or less wavy interfaces. Each lath in
the packet maintains the same variant or
orientation relationship with the parent
crystal. A single grain of the parent phase
can transform into one or more such pack-
ets. The “plate” martensite arrangement
which is observed in the same alloys differs
from the lath configuration, because adja-
cent martensite plates are generally not
parallel to each other.
Diagrams have been constructed for
cases where there is a variety in morphol-
ogy, as for example for Fe–Ni–C alloys.
Maki and Tamura (1987) showed that the
morphology of the a¢-martensite in these
alloys is related to the transformation tem-
perature and the carbon content (Fig. 9-
22).
Distinct martensite plate arrangements
can also be recognized in alloys pertaining
to the b-Hume–Rothery alloys. Schroeder
and Wayman (1977) classified these ar-
rangements into spear, fork, wedge and di-
amond forms. Each representation carries
with it a definite crystallographic relation-
ship between the variants constituting
the arrangement. Grouping of martensite
plates in such arrangements will lead to a
serious reduction in the elastic stresses. By
analysis of the crystallography of the plates
in a single group, it can be shown that the
respective macroscopic shape changes an-
nihilate each other (Tas et al., 1973). Such
group formation is then called “self-ac-
commodation”.
In these b-Hume–Rothery alloys, three
types of martensite form, the 3R-, 9R- and
2H-types. A detailed analysis of the micro-
structure reveals that the martensite vari-
Figure 9-21.Transmission electron micrograph of a
martensite in steel showing the twinned midrib
(courtesy C. M. Wayman, 1989, University of Illi-
nois, Urbana (IL)).www.iran-mavad.com
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670. ants form in six different groups, each
group consisting of four variants. The habit
planes of these four variants are located
around the same {110}
bpole, whereas each
basal plane is located close to one of four
other {110}
bpoles. The total macroscopic
deformation of such a group is to a first ap-
proximation completely compensated. A
more complete reduction in the three-di-
mensional strain is obtained if the total
transformation strain is calculated for the
six groups as an entity. As the martensite
transformation involves a very small vol-
ume change in all b-Hume–Rothery alloys,
the strain accommodation is thus almost
complete.
The shape change associated with a mar-
tensite plate creates stresses in both the
parent and the martensite phases. If these
stresses exceed the flow stress for plastic
deformation, strain accommodation is then
accomplished not only by elastic but also by
plastic deformation in one or both phases.
Partitioning of the parent crystal, with
finer plates forming subsequently in the
partitioned region, frequently occurs and
illustrates the fractal nature of the transfor-
mation (Fig. 9-10). It should be mentioned,
however, that not all martensite micro-
structures show fractal characteristics
(Hornbogen, 1988).
Until now only the more macroscopic
observable features of the microstructure
of martensite have been discussed. Trans-
mission electron microscopy reveals that
the substructure of martensite is also char-
acteristic. It consists, depending on the al-
loy system and alloy composition, of regu-
larly spaced stacking faults (e.g. Cu-base
b¢-type martensite), twins with a constant
thickness ratio (e.g. Fe–30% Ni), disloca-
tions (e.g. Fe–20% Ni–5% Mn), stacking
faults and twins in the same martensite
plate (e.g. Cu–Ga), or twins in the midrib
region surrounded by dislocations.
9.8.1.4 Crystallographic
Phenomenological Theory
The formal phenomenological theories
of martensite formation predict the crystal-
lographic characteristics, such as the shape
deformation, the orientation of the habit
plane, the orientation relationship between
parent and product phase, and the ampli-
tude of lattice invariant deformation. This
prediction is obtained from the sole knowl-
edge of the structures and lattice parame-
ters of the two phases and with the basic as-
sumption that the interface between parent
phase and martensite is undistorted on a
macroscopic scale.
The observation of the K–S and N–W
orientation relationships led us to origi-
nally believe that a martensite was formed
620 9 Diffusionless Transformations
Figure 9-22.Relationship between a¢-martensite
morphology and M
stemperature as a function of car-
bon content in Fe–Ni–C alloys (Maki and Tamura,
1987).www.iran-mavad.com
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671. 9.8 Martensitic Transformations 621
by shear on those planes and directions
specified in the orientation relationships.
However, it was found that the shear mech-
anisms proposed by the K–S and N–W
relations are not consistent with these ex-
perimental observations. The observations
made by Greninger and Troiano (1949) on
Fe–22% – Ni–0.8% C martensite were the
key to the mathematical development of
the crystallographic theory of martensite.
They found that martensite plates exhibited
a surface relief that can be described by a
homogeneous shear along the habit plane,
but this homogeneous shear could not
transform the f.c.c. lattice of the parent
phase into the b.c.t. lattice of the marten-
site. If the f.c.c. lattice had undergone the
same homogeneous deformation, the struc-
ture of the martensite would have been
trigonal. They therefore suggested that two
types of shear are involved in the marten-
sitic transformation: a “first” simple shear
which is responsible for the macroscopic
shape change, and a “second” shear which
needs to be added to obtain the structural
change but which should produce no ob-
servable macroscopic change in shape.
Two years later, Bowles (1951) showed
that the shape deformation may be any in-
variant plane strain. This opened the way to
the formulation of the general theory of the
crystallography by Wechsler et al. (1953)
and, independently, by Bowles and Mack-
enzie (1954). Almost equivalent theories
were later developed by Bullough and
Bilby (1956) and Bilby and Frank (1960).
The reader may consult the following more
elaborate reviews of these theories: Way-
man (1964), Christian (1965), Nishiyama
(1978) and Ahlers (1982).
The basic assumption in the crystallo-
graphic theories is that the interface be-
tween the product and the parent phases is
undistorted, which means that any vector
that lies in this interface on the side of the
martensite would be a vector of the same
size and the same orientation in the parent
phase before transformation. As indicated
in Sec. 9.3.1, the macroscopic shape
change of an invariant-plane transforma-
tion can be represented by a combination
of a pure lattice deformation (B), the so-
called Bain strain, a rigid lattice rotation
(R), and an inhomogeneous lattice-invari-
ant deformation (P). The pure lattice defor-
mation either increases or decreases some
vectors in length. According to Wayman
(1964), “the essence of the crystallographic
theory of martensitic transformations is to
find a simple shear (of a unique amount, on
a certain plane, and in a certain direction)
such that vectors which are increased in
length due to the lattice deformation are
correspondingly decreased in length due to
the simple shear, and vice versa. Such vec-
tors which remain invariant in length to
these operations define the potential habit
plane. Physically speaking, the ellipsoid
generated from the initial sphere by the lat-
tice deformation is distorted by the simple
shear into another ellipsoid which becomes
tangential to the initial sphere, the points of
tangency being related along a diameter.”
This is clearly illustrated in Fig. 9-23,
where the problem becomes two-dimen-
sional, because one of the principal axes of
the lattice deformation is taken as normal
to the plane of shear.
Some complementary remarks concern-
ing the crystallographic theory should be
made. The input data for the calculations
are (i) the lattice parameters of the parent
and martensite phases, (ii) the lattice corre-
spondence, and (iii) the lattice-invariant
shear. The output of the calculations is then
the amount of inhomogeneous shear re-
quired to obtain the invariant plane condi-
tion, the macroscopic shape change, and
the orientation relationship. Because of the
lattice symmetries, differently orientedwww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

672. Brain relations and inhomogeneous shear
systems lead to a number of crystallo-
graphically equivalent solutions. Because
of the observed orientation relationships,
the Bain relationship is fixed for most
cases. However, larger unit cells are some-
times chosen, especially for those marten-
sites where the crystal structure has a large
unit cell compared with that of the parent
phase. The only variable in these calcula-
tions is the choice of the inhomogeneous
shear system. The orientation of the habit
plane is found to be very sensitive to the
choice that is made. For the f.c.c. to b.c.c.
or b.c.t. transformation, the twin shear
(112)
M[111
–
]
Mgives a (3 15 10)
Phabit
plane, whereas a (011)
M[1
–
1
–
1]
Mshear re-
sults in a (111)
Phabit plane.
The phenomenological theory as ex-
plained above is based on one active shear
system. However, for some alloy systems
this is not adequate. For example, a single
(112)
Mtwinning system is not able to ex-
plain the {225}
Phabit plane in some steels,
and even two inhomogeneous shear sys-
tems do not give agreement with the ex-
perimental observations. Similar disagree-
ments have been observed in other alloy
systems. To test critically the validity of
the crystallographic theories, all crystallo-
graphic parameters should be measured
and compared with the theoretical predic-
tions. Agreement should be obtained for
the complete set of parameters. For the
martensites that are twinned, this includes
a careful determination of the normal to the
twinning plane K
1relative to the parent lat-
tice. In Cu–Al–Ni, for example, inconsis-
tencies up to 12.5° have been found.
In cases where the lattice-invariant shear
is twinning (as opposed to faulting or slip),
type I twinning, where the twin plane orig-
inates from a mirror plane in the parent
phase, has been assumed. Otsuka (1986)
carefully analyzed for a number of systems
the possibility of a type II twinning as an
alternative inhomogeneous shear. In type II
twinning, the shear direction stems from a
two-fold symmetry direction of the parent
phase. Otsuka (1986) compiled all the
twinning modes observed in martensite
into a table and found that most of them are
type I or compound but that type II twin-
ning had only recently been observed.
According to Nishida and Li (2000), five
different twinning modes exist in TiNi
and other shape-memory alloys such as
Cu–Al–Ni, Cu–Sn etc., namely the {111}
type I, the {011} type I, the ·011Òtype II,
the (100)-compound and the (001)-com-
pound twins. Type II twinning has recently
received much attention as a mechanism
for lattice-invariant shear in some alloys.
Because type II twins have irrational twin
boundaries, the physical meaning of an ir-
rational boundary is still a controversial
problem. It has been proposed that an irra-
tional boundary consists of rational ledges
and steps, the average being irrational.
Thereafter, Hara et al. (1998) carried out a
careful study using HRTEM, to try and ob-
serve ·111Òtype II twin boundaries in a
622 9 Diffusionless Transformations
Figure 9-23.Production of an undistorted plane by
shear such that the shape ellipsoid touches the unit
sphere along one of its principal axes (Christian,
1965).www.iran-mavad.com
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673. 9.8 Martensitic Transformations 623
Cu–Al–Ni alloy, but they were unable to
observe any ledges or steps. The boundary
is always associated with dark strain con-
trast, and the lattice is continuous through
the irrational boundary. Nishida and Li
(2000) also carried out extensive studies on
·011Òtype II twin boundaries in TiNi using
HRTEM, but they did not observe ledges or
steps either. Based on these experimental
results, it is thus most likely that the type II
thin boundary is irrational even on a micro-
scopic scale, and the strains at the boun-
dary are elastically relaxed with wide twin
width. To confirm this interpretation. Hara
et al. (1998) carried out computer simula-
tions by using the molecular dynamics
method. The result showed that the irra-
tional thin boundary did not show any
steps. Thus, the above interpretation for an
irrational twin boundary is justified. Ot-
suka and Ren (1999) have pointed out
again the importance of type II twinning in
the crystallographic aspects of martensite.
They also stress the role that martensite
aging has on the rubber-like behavior of
martensite, a point that has been a long-
standing unsolved problem. They showed
that the point defects play a fundamental
role. The deformation mechanisms of the
cold deformation of NiTi martensite have
been thoroughly analyzed by Liu et al.
(1999a, b). They also found an interplay
between type I and type II twinning.
As already mentioned in Sec. 9.3.6, a
better and more complete agreement can be
achieved when the strain energy terms,
both bulk and interfacial, are included in
the calculation.
9.8.1.5 Structure of the Habit Plane
In a number of alloys, especially those in
which the so-called thermoelastic marten-
sites are formed, the interface between
martensite and the parent phase is mobile,
even at very low temperatures. This obser-
vation shows that the interface migration
must be accomplished without appreciable
thermal activation. The interface is thus
“glissile”. In searching for models to ex-
plain the structure and mobility of the
interface, we are concerned with the ideal
and the actual interface morphology. A
careful experimental analysis of the inter-
face structure is therefore required if we
want to verify the various models that have
been proposed. As the models treat the
interface on an atomistic scale, the sub-
structure of the interface should be studied
by conventional and by high-resolution
transmission electron microscopy. The
same holds for martensite-to-martensite
interfaces, which in some alloys are also
mobile. Recently, atomistic imaging of the
martensite/austenite and martensite/mar-
tensite interfaces have been obtained. It is
therefore not surprising that both aspects,
the observation of interface substructures
and the atomistic models, are treated
jointly in the literature. For further reading
concerning the interface structures and the
growth mechanism of martensite we refer
to the review papers by Christian (1982),
Christian and Knowles (1982), and Olson
and Cohen (1986). A summary of these pa-
pers is given below.
Let us first introduce the kinds of mar-
tensite interfaces concerned: glissile and
non-glissile martensitic interfaces, with the
latter subdivided into the coherent and
semi-coherent interfaces. The two struc-
tures, martensite and the parent phase, are
said to be “fully coherent” if both lattices
have a matching plane parallel to the inter-
face. If a fully coherent interface is dis-
played, the crystal undergoes a shape de-
formation leaving all vectors in the inter-
face invariant. In general, the two phases
do not have a plane of atomic fit, so that
fully coherent martensite interfaces are ex-www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

674. ceptional. A fully coherent martensitic in-
terface is, for example, that between f.c.c.
and h.c.p. structures with lattice parame-
ters such that a(f.c.c.) =÷
–
2a(h.c.p.); the
atomic arrangement in the basal planes,
which constitute the interface between the
two structures, are identical. Such transfor-
mations are found in Co and its alloys and
in some Fe-based alloys. The situation at
semi-coherent interfaces becomes more
complex. The models predict the presence
of dislocations to correct the mismatch
along the interface. If this coherent inter-
face moves, it is suggested that not all vec-
tors are left invariant and that the move-
ment of dislocations causes shear in the
product phase. Fig. 9-24 shows the slip as-
sociated with the interface dislocations.
Internally twinned martensite has been
reported to show a zig-zag parent–marten-
site interface, as observed by conventional
electron microscopy in, for example,
Ti–Mn and Cu–Al–Ni. Fine parallel stria-
tions have been observed in the interface
between austenite and both the b¢-type and
the g¢-type Cu–Al–Ni-martensite. These
striations have been accounted for in terms
of interfacial dislocations resulting from
random faulting on the basal plane of the
b¢-type martensite and the twinning planes
of the g¢-type martensite. High-resolution
electron micrographs show that the inter-
face between martensite and the parent
phase and also the intervariant interfaces
and the interfaces between the internal
twins in one martensite plate contain dis-
continuities (“steps”) on an atomic scale,
the nature of which has not yet been com-
pletely unravelled. These steps can be con-
sidered as resulting from a small deviation
of the ideal habit plane, and would then be
comparable to those observed along the
interfaces of tapered twins.
An exact understanding of the structure
of the interfaces involved in the martensitic
transformation (the parent–martensite, the
intervariant, and the twin/twin interfaces)
is therefore essential in determining the
mechanism of transformation and the mo-
bility of the interfaces.
9.8.2 Thermodynamics and Kinetics
of the Martensitic Transformation
9.8.2.1 Critical Driving Force
and Transformation Temperatures
A quantitative thermodynamic treatment
of the martensitic transformation requires a
precise knowledge of the thermodynamic
equilibrium temperature T
0and of the
change in Gibbs energy at the transforma-
624 9 Diffusionless Transformations
Figure 9-24.Three-dimen-
sional representation of a
semi-coherent martensite
interface; the vectors OA are
distorted into O¢A¢but the
large vectors OZ = O¢Z ¢are
invariant (Christian, 1982).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

675. 9.8 Martensitic Transformations 625
tion temperature M
s. Both can be
calculated and/or derived from measured
data, as is shown here for two examples:
the martensitic transformation in Fe–X
(X = Ni, Ru, …) and in Cu–Zn–Al alloys.
In the former example, both the parent and
the martensite phases have the same struc-
ture as the equilibrium phases and hence
the data for the equilibrium phases can be
taken. In the latter example, both structures
differ from that of the equilibrium phases,
which requires a more elaborate calculation.
For the Fe–X alloys, the Gibbs energy
per mole of the parent austenite phase g
and of the martensite phase aare G
g
and
G
a
, respectively. The change in Gibbs en-
ergy per mole, DG
gÆa
, which for a mar-
tensitic transformation gÆais available
to the system at any temperature T, is then
DG
gÆa
|
T= G
a
– G
g
(9-8)
This quantity is negative for temperatures
at which the a-phase is the more stable and
positive for temperatures at which the g-
phase is the more stable. There is a charac-
teristic temperature T
0corresponding to the
thermodynamic equilibrium between both
phases, such that
DG
gÆa
|
T=T0
= 0 (9-9)
Because the transformation creates interfa-
cial and elastic energies, the martensitic
transformation gÆaor aÆgdoes not
start at T
0, but at a temperature below or
above T
0, respectively. It is therefore nec-
essary to undercool or overheat, respec-
tively, until M
sor A
sis reached. At these
temperatures the Gibbs energy change
DG
gÆa
is sufficiently large to induce the
forward or reverse transformation, respec-
tively. DG
gÆa
(at M
s) is then the critical
chemical driving force.
The martensite phase, represented by M,
is to be regarded as the a-phase embedded
in the g-phase. Because of the shape and
volume changes associated with the trans-
formation, elastic strain energy also has to
be considered. The Gibbs energy is thus
composed of chemical Gibbs energy, G
c,
and strain energy, E
e, so that the Gibbs en-
ergy change accompanying the transforma-
tion may be written as
G
gÆM
= DG
c
gÆa+ DE
e
aÆM (9-10)
At temperatures below M
s, where both
phases coexist and thus are in equilibrium,
DG
gÆM
|
T=0. DG
c
gÆa|
Tis then exactly
equal, but opposite in sign, to the sum of all
non-chemical energies DG
nc
aÆM|
T. If the
surface energies are neglected in compari-
son with the high strain energies, the non-
chemical energy equals DE
e
aÆM|
T, and
approaches zero at T=M
s. The strain en-
ergy stored in the material is the sum of
that produced by shearing and by volume
change. The former depends on the
strength of the parent phase and thus also
on the grain size, hence M
salso depends on
the grain size, as shown by Hsu and Xiao-
wang (1989).
The necessary undercooling (T
0–M
s)
and superheating (A
s–T
0) vary for differ-
ent alloy systems, and for certain materials
even with composition. A precise thermo-
dynamic definition of M
sand A
scannot be
given, however, if the non-chemical ener-
gies, DG
nc
aÆM, are not known. We can then
only say that M
sor A
sis the temperature at
which the quantity DG
gÆa
(at T=M
sor
T=A
s, respectively) is sufficiently nega-
tive or positive, respectively, to have a rea-
sonable chance of nucleation.
Two approaches are found in the litera-
ture for calculating the critical chemical
driving force. The first is based on the ex-
perimentally determined M
stemperatures
(Kaufman, 1965) and the other on a theo-
retical model for the non-chemical energies
(Hsu, 1985).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

676. The first approach has been used for fer-
rous alloys, which can be classified into
two systems: those with g-loops and those
with stabilized g -phases (see Fig. 9-25).
Figure 9-25 also gives the M
stemperatures
for the a- and e-martensite. If Ais the al-
loying element for iron, the molar chemical
Gibbs energy for the austenite phase (G
g
)
can be written as
G
g
= (1 – x )G
g
Fe
+ xG
g
A
+ G
g
m
(9-11)
where xrepresents the atomic fraction of
the element A in solid solution in the g-
austenite, (1 –x) the atomic fraction of
iron, G
g
Fe
the chemical Gibbs energy of
pure iron as f.c.c. g -phase, G
g
A
the chemical
Gibbs energy of element A as f.c.c. phase,
and G
g
m
the Gibbs energy of mixing of the
g-phase. Similarly, the Gibbs energy of the
a-phase can be given as
G
a
= (1 – x)G
a
Fe
+ xG
a
A
+ G
a
m
(9-12)
where G
a
Fe
and G
a
A
are the Gibbs energies of
pure iron and pure element A as a b.c.c. a-
phase, respectively, and G
a
m
is the Gibbs
energy of mixing of the martensite phase.
The change in chemical Gibbs energy ac-
companying the martensitic transformation
gÆathen becomes
DG
gÆa
= (1 – x )DG
Fe
gÆa+ xDG
A
gÆa
+ DG
m
gÆa (9-13)
The quantity DG
Fe
gÆarepresents the Gibbs
energy change for transformation gÆaof
pure iron and can be assessed experimen-
tally from the measured heat of transforma-
tion and the specific heat of both phases.
The quantity DG
A
gÆacannot usually be ob-
tained from experiments because the ele-
ment A does not always exist in the two
modifications gand a; it must therefore be
estimated from thermodynamic models for
solid solutions. The quantity DG
m
gÆais the
626 9 Diffusionless Transformations
Figure 9-25.Schematic dia-
grams for ferrous alloys that
form a g-loop (Fe–Cu, Cr,
Mo, Sn, V, W) and that g-
loop forms a stabilized aus-
tenite phase (Fe–C, Ir, Mn,
N, Ni, Pt, Ru); (a) equilib-
rium diagrams; (b) M
stem-
perature diagrams (Krauss
and Marder, 1971).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

677. 9.8 Martensitic Transformations 627
difference in Gibbs energy of mixing and
can in principle be measured experimen-
tally through activity measurements; if not,
it must also be estimated.
The equilibrium temperature T
0and the
critical driving force and M
sand A
scan be
calculated from Eq. (9-12). Such calcula-
tions have been performed by Kaufman
(1965) for the iron–ruthenium alloy sys-
tem, which is of particular interest because
the g-phase transforms martensitically into
two phases, the a-b.c.c. and the e-hexago-
nal phases. Both phases also occur as equi-
librium phases, as shown in Fig. 9-26 to-
gether with observed M
sand A
stempera-
tures and the calculated T
0temperatures.
The undercooling for the a-martensite
formation is strongly composition depen-
dent, whereas it is independent of composi-
tion for the hexagonal martensite. The
computed T
0curves are seen to lie between
the appropriate transformation temperature
curves. The calculated driving forces,
DG
gÆa
and DG
gÆe
, for both transforma-
tions are plotted as a function of temperature
for various compositions. The intersections
of these curves with the temperature axis
correspond to the theoretically deduced T
0
temperatures for the appropriate composi-
tions. When the appropriate experimentally
derived M
sand A
sare cross-plotted, the crit-
ical driving forces for the g/aand g/emar-
tensitic transformations are obtained. The
latter is seen to be smaller, which is consis-
tent with the closer lattice correspondence
of the former transformation.
Hsu (1985) presented a model by which
more accurate computations of the non-
chemical part DG
aÆM
are possible for
Fe–C, Fe–X and Fe–C–X alloys. This en-
abled him to obtain from Eq. (9-9) the
theoretical M
stemperatures, which are in
good agreement with the observed values.
In the second example, martensite for-
mation in Cu–Zn alloys, the change in
Figure 9-26.(a) The iron–ruthenium phase diagram
and (b) the T
0and M
sand A
stemperature diagrams
(after Kaufman, 1965).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

678. Gibbs energy can, according to Hsu and
Zhou (1989), be described as
DG
b¢–M
= DG
b¢–b
+ DG
b–a
+ DG
a–a¢
+ DG
a¢–M
(9-14)
where b¢– M represents the transformation
from the the ordered b.c.c. phase to the or-
dered 9R-type of martensite, b¢–bthe or-
der–disorder transition, b–athe transfor-
mation from the disordered b.c.c. to the
disordered f.c.c. phase having the same
composition, a–a¢the disorder–order
transformation in the f.c.c. phase, and
a¢– M the transition from the ordered f.c.c.
phase to the ordered martensite phase. As-
suming a simplified relationship between
the degree of ordering and temperature,
Hsu and Zhou found good agreement
between the calculated and observed M
s.
Their calculations show that ordering of
the parent phase, which cannot be sup-
pressed even by severe quenching, strongly
influences T
0.
It is known that martensite may also be
induced by an external stress at tempera-
tures above M
s. The problem now is to cal-
culate the change in T
0due to changes in
stress. As a first approximation, it is as-
sumed that the driving force DG
m
PÆM|
T=M s
required for nucleation remains constant
with temperature and thus independent of
stress. Patel and Cohen (1953) calculated
the work done on the stressed specimen;
their treatment provides a good under-
standing of how an applied stress that is de-
composed into a shear stress along the
habit plane and a normal stress perpendicu-
lar to it, affects the transformation temper-
ature. At M
s
s
, which is the martensite start
temperature when cooling under an applied
stress
s, the chemical Gibbs energy change
equals the transformation work of the ex-
ternal stress:
DG
s
PÆM
= 1/2s
a (9-15)
¥[
d
0sin 2q±e
0(1 + cos 2q)]V
m
where d
0is the shear strain, s
athe applied
stress,
qthe angle between the stress axis
and the normal to the operative shear plane,
e
0the corresponding strain associated with
the transformation, and V
mthe molar vol-
ume. The quantity D H
PÆM
can be meas-
ured by calorimetry and DS
PÆM
can be
evaluated from stress-induced transfor-
mation experiments or calculated from Eq.
(9-16). The temperature T
0can be calcu-
lated thermodynamically or obtained more
or less accurately from the relationships
DG
PÆM
= DH
PÆM
– TDS
PÆM
DH
PÆM
= T
0DS
PÆM (9-16)
and
T
0= (A
s– M
s)/2 = (A
f– M
f)/2 (9-17)
However, it should be noted that the deter-
mination of T
0does not always obey these
simple relationships and that the calorimet-
628 9 Diffusionless Transformations
Figure 9-27.Gibbs energy G* versus temperature
and force for stressed samples: P and M represent
free energy surfaces for parent and martensite, re-
spectively (Wollants et al., 1979).www.iran-mavad.com
+ s e l ≤'4 , kp e r i ≤&s ! 9 j+ N 0 e

679. 9.8 Martensitic Transformations 629
rically measured heats do not always re-
flect the exact heats of transformation.
As stress itself is also a state variable in-
dependent of temperature, it should be con-
sidered in the thermodynamic treatment as
explained by Wollants et al. (1979). To de-
scribe the thermodynamic state of a uniax-
ially stressed crystal, they introduced the
“elastic” state functions H* and G*, which
incorporate the effect of stress as follows:
H* = U + PV– Fl= H– Fl
= H–
seV
m (9-18)
G* = U + PV– TS– Fl= G– Fl
= G–
seV
m (9-19)
where Fis the applied force and lthe “mo-
lar length” of the crystal. Fig. 9-27 illus-
trates how the equilibrium temperature and
force change when one of the variables is
changed; P and M represent the Gibbs energy
surfaces of the parent phase and of marten-
site, respectively. At the two-phase equilib-
rium G*
P
=G*
M
and if, at constant hydro-
static pressure, the intensive variables F
(or
s) and Tare changed in such a way
that there is thermodynamic equilibrium
between martensite and the parent phase,
then dG*
P
=dG*
M
, or
–S
P
dT– l
P
dF= – S
M
dTl
M
dF
so that dF/dT=–[DS/Dl]
PÆM
, or, since
the molar work FDl
PÆM
=se
PÆM
V
mand
DS
PÆM
=DH*(s)/T
0(s), it also follows
that
(9-20)
d
s/dT= – [DH*( s)]/[T
0(s)e
PÆM
V
m]
where V
mis the molar volume (V
m=V
P
=
V
M
). Eq. (9-20) is the Clausius–Clapeyron
equation for a uniaxial stress, which is sim-
ilar in form to that for hydrostatic pressure,
except for the negative sign.
The change in critical stress necessary to
induce martensite can be obtained from
tensile tests carried out at different temper-
atures (see Fig. 9-28). The elongation re-
sulting from the transformation is orienta-
tion dependent (Fig. 9-29). From data such
as those shown in Figs. 9-28 and 9-29, we
can calculate DS
P–M
. It is evident that for
each crystal orientation the slope d
s/dTis
different.
Figure 9-28.Results of tensile tests for inducing
martensite in a Cu–34.1 Zn–1.8 Sn (at.%) alloy
(Pops, 1970).
Figure 9-29.Orientation dependence of stress–
strain curves for martensite formation in a Cu–Al–Ni alloy (Horikawa et al., 1988).www.iran-mavad.com
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680. Similarly, martensite can be induced by
magnetic fields. By taking into account the
composition, the influence of grain boun-
daries, and crystal orientation, the mag-
netic invar effect and the austenite magne-
tism, Shimizu and Kakeshita (1989) pro-
posed an equation that describes the shift
of M
sas a function of the magnetic field.
9.8.2.2 Nucleation
Martensitic transformations are first-or-
der phase transformations and hence occur
by nucleation and growth. In most in-
stances, except in the case of thermoelas-
ticity (see below), the growth of a marten-
site plate proceeds so rapidly that the trans-
formation kinetics are dominated by the
nucleation event. Various mechanisms of
martensite nucleation have in the past been
proposed and can be considered under two
subheadings. In the first group of models,
the so-called localized nucleation models,
concepts of diffusional nucleation kinetics
are applied; the second group of models is
based on lattice instability considerations
concerning both static and dynamic lattice
instability. All nucleation models can fur-
ther be divided into classical and non-clas-
sical. The former model involves lattice
perturbations of fixed amplitude and vary-
ing size, whereas the latter considers per-
turbations of varying size (Olson and Co-
hen, 1982b).
In the classical nucleation theory; mar-
tensite nuclei form along a path of constant
composition and structure and the state of
the nucleus is given by its size. Because the
martensitic transformation involves shear
strains, it can be shown that the strain en-
ergy is minimized for a disc-like nucleus,
but then the surface energy becomes very
large. The critical nucleus, assuming an
oblate spheroidal shape (Fig. 9-30), will
then have an aspect ratio (c/r) such that for
any change in shape, the decrease in strain
energy will be exactly balanced by an in-
crease in interfacial energy. The interfacial
Gibbs energy per plate is
vDg
s= 2pr
2
G (9-21)
where vis the volume of the plate, Dg
sthe
surface Gibbs energy per unit volume, and
Gthe interfacial energy. The strain energy
per plate is
vDg
e= (4/3)pr
2
c(Ac/r) (9-22)
where Dg
e(=Ac/r) is the strain energy per
unit volume and Ais a factor to be deduced
from linear elasticity and thus a function of
the elastic constants and of the shear and
dilatational strains. The chemical Gibbs
energy change per plate is
vDg
cor (4/3)pr
2
cDg
c (9-23)
If the nucleation occurs at a lattice defect,
we have to consider also the Gibbs energy
G
ddue to the defect and the nucleus–de-
fect interaction energy G
i. According to
Olson and Cohen (1982), the total Gibbs
energy describing the formation of a classi-
630 9 Diffusionless Transformations
Figure 9-30.Shape of a nucleus of a martensite
plate.www.iran-mavad.com
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681. 9.8 Martensitic Transformations 631
cal martensitic nucleus becomes
(9-24)
G(r,c) = G
d+ G
i+ v(Dg
c+ Dg
e+ Dg
s)
and is given schematically in Fig. 9-31.
Three cases are considered in calculating
the critical free energy for nucleation DG*
and the critical nucleus size r* and c*.
In the case of homogeneous nucleation,
G
dand G
iare zero, and on inserting the
necessary quantities into Eq. (9-24) we
find that the barrier DG* is too high by sev-
eral orders of magnitude. Even assuming
local compositional fluctuations or the ex-
istence of pre-existing embryos does not
give full satisfaction. It was therefore soon
recognized that homogeneous nucleation
of martensite is impossible. Recently,
much progress has been made in the under-
standing of nucleus formation at lattice de-
fects. DG* and therefore also the critical
size of the nucleus can be reduced by as-
suming the nucleation at a defect. Under
certain special conditions, this heterogene-
ous nucleation may even be barrierless.
Such a case applies to the f.c.c.-to-h.c.p.
transformation, which may take place by
dissociation of a number of properly
spaced total dislocations present in the ma-
trix phase into partial dislocations separ-
ated by stacking faults. The stacking-fault
Figure 9-31.Schematic
nucleus Gibbs energy (G )
curves for nucleation via a
classical path: (a) homoge-
neous, (b) heterogeneous,
and (c) barrierless nuclea-
tion (Olson and Cohen,
1982b).
Figure 9-32.Electron mi-
crographs of the nucleation and early growth stage around inclusion particles in a Ti–Ni–Cu alloy (Saburi and Nenno, 1987).www.iran-mavad.com
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682. energy is temperature dependent and be-
comes positive below T
0, which results in a
barrierless nucleation.
In a number of alloy systems, softening
of certain elastic constants is observed and
it is then argued that, although the homoge-
neous soft-mode concept is definitely not
adequate to describe the nucleation of mar-
tensite, stresses and strains present around
defects of the lattice can induce a local me-
chanical instability. Such a model is called
“the localized soft-mode concept” (Guénin
and Clapp, 1986). In this model the lattice
Gibbs energy is a function of pure strains
and therefore of second- and third-order
elastic constants. The third-order constants
(which relate the strain energy to the
amount of strain) introduce anharmonic
terms into the strain energy and may lead to
mechanical instability. The region of me-
chanical instability, or the “strain spino-
dal”, is so defined that any further increase
in strain will make the lattice unstable with
respect to a decomposition into strained re-
gions. In these zones a nucleus can develop
without generating any strain energy, and
the only resisting term remains the surface
energy. This results in a reduced critical
size of the nucleus, which is further de-
creased as the temperature is lowered ow-
ing to the increase in chemical driving
force.
In situelectron microscope observations
have been made on the nucleation and early
stages of growth of martensite, as shown in
Fig. 9-32. Martensite nucleates at stress
concentrations, the nucleation takes place
repeatedly at the same place, and the strain
contrast disappears as nucleation and
growth proceed and reappears when mar-
tensite disappears.
At present, the nucleation models are be-
ing further refined by molecular dynamic
calculations.
9.8.2.3 Growth and Kinetics
A distinction is made between the kinet-
ics of a single martensite plate and the glo-
bal kinetics, which expresses the volume
fraction of the parent phase that is trans-
formed. According to the observed kinet-
ics, martensitic transformations can be di-
vided into two distinct classes: athermal
and isothermal martensite. In athermal mar-
tensite, the transformation progresses with
decreasing temperature, whereas in iso-
thermal martensite, the transformation pro-
gresses with time at a constant temperature.
The growth may be “thermoelastic” or of
the “burst” type. The latter is the more
common mode. It consists of the formation
of comparatively large amounts of marten-
site (typically 10–30 vol.%) in “bursts”
that are caused by autocatalytic nucleation
and rapid growth of numerous plates. Each
individual martensite plate is completely
formed with a speed higher than 10
5
cm/s
and the transformation progresses by the
formation of new plates. The global kinet-
ics of the transformation are therefore es-
sentially controlled by the nucleation fre-
quency. The thermoelastic growth mode is
characterized by the formation of thin, par-
allel-sided plates or wedge-shaped pairs of
plates (Fig. 9-33), which form and grow
progressively as the temperature is lowered
below M
sand which shrink and disappear
on reversing the temperature change. This
behavior arises because the matrix accom-
modates the shape deformation of the mar-
tensite plate elastically, so that at a speci-
fied temperature the transformation front
of the plate and the matrix are in thermody-
namic equilibrium. Any change in temper-
ature displaces this equilibrium and, there-
fore, the plate grows or shrinks. A com-
plete mechanical analog of this thermoelas-
tic behavior is the pseudoelastic behavior.
The growth or shrinkage of individual mar-
632 9 Diffusionless Transformationswww.iran-mavad.com
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683. 9.8 Martensitic Transformations 633
Figure 9-33.Thermoelastic behavior in Ag–Cd alloys showing the
growth of self-accommodating groups of martensite plates (Delaey
et al., 1974).
Heating Cooling
Figure 9-34.Schematic representation of some relevant features (volume-transformed product or transforma-
tion strain) experimentally observed in hysteresis curves corresponding to thermally induced and stress-induced thermoelastic transformations: (a, e) single interface transformation in a single crystal; (b, f) multiple interface transformation; (c, g) discontinuous jumps (bursts), (d, h) partial cycling behavior.www.iran-mavad.com
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684. tensite plates is then a direct function of the
increase or decrease in stress. More elab-
orate thermodynamic treatments of ther-
moelasticity can be found in the papers by
Delaey et al. (1974), Salzbrenner and Co-
hen (1979), Ling and Owen (1981) and Or-
tin and Planes (1989).
The best quantitative understanding of
the kinetics of the martensitic transforma-
tion is obtained from isothermal transfor-
mations, because they permit both the nu-
cleation and the transformation rates to be
determined. In those alloys exhibiting iso-
thermal martensite (Thadhani and Meyers,
1986), it is shown that at each temperature
the transformation starts in the austenite
and proceeds as a function of time. The
transformation exhibits a C-curve behav-
ior. Isothermal martensitic transformation
kinetics consist of two effects: an initial in-
crease in the total volume fraction of mar-
tensite, which is attributed to an autocata-
lytic nucleation of new martensite plates,
followed by a decrease due to the compart-
mentalization of the austenite into smaller
and smaller areas.
9.8.2.4 Transformation Hysteresis
Hysteresis behavior is one of the pecu-
liar characteristics of both the thermal and
stress-induced martensitic transformations.
In several studies the origin of the fric-
tional resistance opposing the interfacial
motion of martensite plates has been inves-
tigated and described. From a practical
point of view, the hysteresis phenomenon
is an important problem in the application
of shape-memory alloys. In general hyster-
esis appears when, on passing through a lo-
cal extreme value (maximum or minimum)
of any control parameter such as tempera-
ture or stress, one or more state variables
do not follow the original path in state
space. When all the state variables, includ-
ing the control parameter, return to their
original values, a closed loop is formed
(Fig. 9-34). The loop is always contoured in
such a sense that it encloses a positive area,
representing the energy lost in the cyclic
process. Therefore, hysteretic behavior is
always related to an energy-dissipative pro-
cess. The dissipated energy is much smaller
in thermoelastic martensitic transforma-
tions than in burst-type transformations.
9.9 Materials
9.9.1 Metallic Materials
A classification of the diffusionless dis-
placive transformations in metallic materi-
als is given in Table 9-4, where the alloy
systems are subdivided into three groups.
The origin of the martensitic transforma-
tion in the first group lies in the allotropic
transformation of the pure solvent. The
parent phase of the alloys of this group thus
does not show any remarkable mechanical
instability. The second group consists of
the
bb.c.c. Hume–Rothery alloys, which
are characterized by a moderate lattice in-
stability in the temperature range above
M
s. The third group is characterized by a
drastic mechanical instability of the parent
phase. Because the transformation is only
weakly first order (by this we mean a dis-
continuous jump in the corresponding ther-
modynamic property whose height, how-
ever, is very small) or even second order, it
is in this group of alloy systems that we
find, in addition to the martensitic, the
quasi-martensitic transformations.
Traditionally, the ferrous and non-fer-
rous martensites have been treated separ-
ately in the literature. Before going into de-
tail it will be an advantage to first compare
and contrast ferrous and non-ferrous mar-
tensites and to do it in such a way that we
634 9 Diffusionless Transformationswww.iran-mavad.com
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685. 9.9 Materials 635
can establish criteria to differentiate the
two groups. Although not every detail of
transformation behavior will be discussed,
a set of criteria have been chosen as shown
in Table 9-5. A more detailed review of
martensite in metallic systems is given by
Nishiyama (1978).
9.9.1.1 Ferrous Alloys
Martensitic transformations in ferrous
alloys have been studied extensively, espe-
cially the crystallography and morphology
which have been reviewed by Muddle
(1982). Depending on alloy composition, a
distinction is made among the various mar-
tensites, based either on the crystallogra-
phy, morphology or growth characteristics.
Essentially, three different crystal struc-
tures appear: the b.c.c. or b.c.t. a¢-marten-
site, the h.c.p. e-martensite, and the long-
range ordered f.c.t. martensite. In plain car-
bon steels martensite is regarded as a
supersaturated, interstitial solid solution of
carbon in b.c.c. iron (ferrite), with a crystal
structure that is a tetragonally distorted
version of the ferrite structure. The tetrago-
nality is linearly dependent on the carbon
Table 9-4.Classification of metallic alloy systems showing diffusionless displacive transformations (Delaey
et al., (1982).
1. Martensite based on allotropic transformation of solvent atom
1. Iron and iron-based alloys
2. Shear transformation, close packed to close packed
1. Cobalt and alloys f.c.c. Æh.c.p., 126 R SF*
2. Rare earth and alloys f.c.c., h.c.p., d.h.c.p., 9 R
(3. MnSi, TiCr
2 NaClÆNiAs, Laves)
3. Body centered cubic to close packed
1. Titanium, zirconium and alloys b.c.c.Æh.c.p., orth. f.c.c tw, d*
2. Alkali and alloys (Li) b.c.c.Æh.c.p.
3. Thallium b.c.c.Æh.c.p.
4. Others: plutonium, uranium, mercury, Complex structures
etc. and alloys
2.b-b.c.c. Hume–Rothery and Ni-based martensitic shape-memory alloys
1. Copper-, silver-, gold-, b-alloys
(disord., ord.) b.c.c. AB, ABABCBCAC, ABAC
2. Ni–Ti–X b-alloys b.c.c. Æ9 R, AB tw, SF*
Nickel b -alloys (Ni–Al) b.c.c. ÆABC tw, SF*
Ni
3–xM
xSn (M = Cu, Mn) b.c.c.ÆAB tw*
(Cobalt b-alloys, Ni–Co–X)
3. Cubic to tetragonal, stress-relaxation twinning or martensite
1. Indium-based alloys f.c.c. Æf.c.t., orth. tw, tw
s
*
2. Manganese-based alloys f.c.c. Æf.c.t., orth. tw
s
*
3. A 15 compounds, LaAg
xIn
1–x b-W.Ætetr.
4. Others: Ru–Ta, Ru–Nb, YCu, LaCd
* SF: stacking faults; tw
s
: (stress relaxation) twins; d: dislocatedwww.iran-mavad.com
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686. content. This a¢-martensite is also found in
a number of substitutional ferrous alloys,
the martensite being either b.c.c. or b.c.t. In
certain alloy systems with an austenite
phase of low stacking fault energy, a mar-
tensitic transformation to a fully coherent
h.c.p. product (e-martensite) is observed.
In some long-range ordered alloys, as in
Fe–Pt and Fe–Pd, f.c.t. in addition to b.c.t.
martensite is observed.
As far as the morphology is concerned,
plate, lath, butterfly, lenticular, banded,
thin-plate and needle-like martensite can
be distinguished.
636 9 Diffusionless Transformations
Table 9-5.A qualitative comparison between ferrous and non-ferrous martensites (Delaey et al., 1982b).
Ferrous martensite Non-ferrous martensite
Interstitial and/or substitutional Nature of alloying Substitutional
Martensitic state in interstitial Hardness Martensitic state is not much
ferrous alloys is much harder harder and may even be softer
than the austenite state than the austenite state
Large Transformation hysteresis Small to very small
Relatively large Transformation strain Relatively small
High values near the M
s Elastic constants of Low values near the M
s
the parent phase
Negative near the M
s Temperature coefficient Positive near the M
s
in most cases of elastic shear constant in many cases
Self-accommodation is not obvious Growth character Well developed self-accommodating
variants
High rate, “burst”, athermal Kinetics Slower rate, no “burst”,
and/or isothermal transformation no isothermal transformation,
thermoelastic balance
High Transformation enthalpy Low to very low
Large Transformation entropy Small
Large Chemical driving force Small
No single interface Growth front Single interface possible
transformation observed
Low and non-reversible Interface mobility High and reversible
Low Damping capacity High
of martensitewww.iran-mavad.com
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687. 9.9 Materials 637
A distinct substructure, crystallographic
orientation of the habit plane and austenite
to martensite orientation relationship are
associated with each morphology, as sum-
marized in Table 9-6.
Because the carbon atom occupies octa-
hedral interstices in the austenite f.c.c. lat-
tice, special attention is drawn to the
Fe–X–C martensite. In the martensite lat-
tice, those interstitial positions are defined
by the Bain correspondence (Fig. 9-35).
Only those at the midpoints of cell edges
parallel to [001]
Band at the centers of the
faces normal to [001]
Bare permitted. This
preferred occupancy affords an explanation
of the observed tetragonality c/a, the de-
gree of which is a function of the carbon
content:
c/a= 1 + 0.045 (wt.% C) (9-25)
Careful X-ray diffraction of martensite,
freshly quenched and maintained at liquid
nitrogen temperature, has shown signifi-
cant deviations from the above equation.
The tetragonality is abnormally lower for
X = Mn or Re and abnormally higher for
X = Al or Ni. Heating to room temperature
of the latter martensite results in a lowering
of the tetragonality. The formation of do-
mains or microtwinning in the former al-
loys and ordering of the Al atoms in the lat-
ter have been put forward as the origin of
the abnormal c/aratio. This behavior has
moreover been related to the martensite
plate morphology (Kajiwara et al., 1986,
1991). Kajiwara and Kikuchi found that in
Fe–Ni–C alloys the tetragonality is abnor-
mally large and depends on the microstruc-
ture. It is very large for a plate martensite,
while it is normal or not so large for a len-
ticular martensite. They conclude that “the
martensite tetragonality is dependent on
the mode of the lattice deformation in the
martensitic transformation. If the lattice
deformation is twinning, the resulting c:a
is large, while in the case of slip it is small”
(Kajiwara and Kikuchi, 1991).
9.9.1.2 Non-Ferrous Alloys
A classification base of the non-ferrous
alloy systems exhibiting martensite is
Table 9-6.Summary of substructure, habit plane
(H.P.) and orientation relationship (O.R.) for the four
types of a¢-martensite (Maki and Tamura, 1987).
Morphology Substructure H.P. O.R.* M
s
Lath (Tangled) (111)
AK–S High
dislocations
Butterfly (Straight) (225)
AK–S
dislocations
and
twins
Lenticular (Straight) (259)
A N
dislocations
and or or
twins (3 10 15)
A
(Mid-rib) G–T
Thin-plate Twins (2 10 15)
AG–T Low
* K–S: Kurdjumow–Sachs relationship, N: Nishiyama
relationship, G–T: Greninger–Troiano relationship
Figure 9-35.Schematic representation of the Bain
correspondence for the f.c.c. to b.c.t. transformation.
The square symbols represents the possible occupied
positions of the interstitial carbon atoms (Muddle,
1982).www.iran-mavad.com
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688. given in Table 9-4, while the alloying ele-
ments are given in Table 9-7. A review of
the non-ferrous martensites was given by
Delaey et al. (1982a).
Two typical examples of the first group
are the cobalt and titanium alloys. The
structure of the cobalt-based martensites
is in general hexagonal close-packed, but
more complex close-packed layered struc-
tures have been reported, such as the 126R,
84R and 48R structures observed in
Co–Al alloys. Because the transformation
is a result of an f.c.c. to h.c.p. transforma-
tion, the basal planes of the martensite
phase are parallel with the (111) planes of
the parent phase and constitute the habit
plane. The structure of the martensite in Ti-
based alloys is also hexagonal but that of
the high-temperature phase is b.c.c. Both
plate and lath morphologies are encoun-
tered in titanium, and also in the similar
zirconium-based alloys. Slip is suggested
as the lattice-invariant deformation mode
in lath martensite, whereas the twining
mode is observed in the plate martensities.
A typical example of the second group
are the copper-, silver- and gold-based
alloys, which have been extensively re-
viewed by Warlimont and Delaey (1974).
Depending on the composition, three types
of close-packed martensite are formed
from the disordered or ordered high-tem-
perature b.c.c. phase, either by quenching
or by stressing. The factors determining the
exact structure are the stacking sequence of
the close-packed structure, the long-range
order of the martensite as derived from the
parent b-phase ordering, and the deviations
from the regular hexagonal arrangements
of the martensite. The last factor is due to
differences in the sizes of the constituent
atoms. The stacking sequence of the main
three phases are ABC, ABCBCACAB and
AB, respectively.
One of the interesting findings is the suc-
cessive stress-induced martensitic transfor-
mations in some of the b-phase alloys dis-
cussed above, as shown clearly if we plot
the critical stresses needed for the transfor-
mation (Fig. 9-36). Stressing a single crys-
tal of Cu–Al–Ni at, for example, 320 K
we find the parent b
1-to-b¢
1, the b¢
1–g¢
1,
638 9 Diffusionless Transformations
Figure 9-36.Critical stresses as a function of tem-
perature for the various stress-induced martensite
transformations in a Cu–Al–Ni alloy (Otsuka and
Shimizu, 1986).
Table 9-7.Schematic representation of some non-
ferrous martensitic alloy systems; the Co-, Ti- and
Zr-based terminal solid solutions, the intermetallic
Ni-, Cu-, Ag- and Alu-based alloys, the antiferro-
magnetic Mn-based alloys and the In-based alloys
(Delaey et al., 1982).www.iran-mavad.com
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689. 9.9 Materials 639
the g¢
1–b≤
1, and finally the b≤
1–a¢
1marten-
site.
The other typical example of the second
group is the Ni–Ti-based alloy system;
both prototype alloy systems constitute the
shape-memory alloys (SMA). The occur-
rence of a so-called “pre-martensitic” R-
shape has long obscured the observations.
The review by Wayman (1987), illustrates
the complexity of the transformation be-
havior. During cooling, the high-tempera-
ture ordered b.c.c. phase (P) transforms
first to an incommensurate phase (I) and on
further cooling to a commensurate phase
(C), and finally to martensite. The P-to-I
transformation is second order, whereas the
I-to-C transformation is a first-order phase
transformation involving a cubic-to-rhom-
bohedral (the so-called R-phase) structural
change. At still lower temperatures the
rhombohedral R-phase transforms into a
monoclinically distorted martensite. The
R-phase also forms displacively and can be
stress-induced, and shows all the character-
istics of a reversible transformation.
Concerning the third group, in only a
few cases, as in In–Tl, has definite proof
been provided to justify the conclusion that
the transformation is martensitic. Most of
the transformations in these systems have
to be classified as quasi-martensitic.
9.9.2 Non-Metals
Inorganic compounds exhibit a variety
of crystal structures owing to their diverse
Table 9-8.Non-metals with lattice deformational transformations (Kriven, 1982).
Inorganic compounds
Alkali and ammonium halides MX, NH
4X (NaCl-cubic¤CsCl-cubic)
Nitrates RbNO
3 (NaCl-cubic¤rhombohedral¤CsCl-cubic)
KNO
3, TlNO
3, AgNO
3 (Orthorhombic¤rhombohedral)
Sulfides MnS (Zinc-blende-type¤NaCl-cubic)
(Wurtzite-type¤NaCl-cubic)
ZnS (Zinc-blende-type¤wurtzite-type)
BaS (NaCl-type¤CsCl-type)
Minerals
Pyroxene chain silicates Enstatite (MgSiO
3) (Orthorhombic ¤monoclinic)
Wollastonite (CaSiO
3) (Monoclinic¤triclinic)
Ferrosilite (FeSiO
3) (Orthorhombic ¤monoclinic)
Silica Quartz (Trigonal¤hexagonal)
Tridymite (Hexagonal, wurtzite-related)
Cristobalite (Cubic ¤tetragonal, zinc blende-related)
Ceramics
Boron nitride BN (Wurtzite type ¤graphite-type)
Carbon C (Wurtzite type ¤graphite)
Zirconia ZrO
2 (Tetragonal¤monoclinic)
Organics
Chain polymers Polyethylene (CH
2–CH
2)
n (Orthorhombic¤monoclinic)
Cement
Belite 2 CaO · SiO
2 (Trigonal¤orthorhombic¤monoclinic)www.iran-mavad.com
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690. chemistry and bonding. Compared with
metals, the relatively low-symmetry parent
structures have fewer degrees of freedom
on transforming to even lower symmetry
product structures, or vice versa. Many of
these transformations involve changes in
electronic states with relatively small volume
changes. They tend to proceed by shuffle-
dominated mechanisms. However, shear
transformations involving large structural
changes in terms of coordination number
or volume changes have also been reported
in inorganic and organic compounds, min-
erals, ceramics, organic compounds, and
some crystalline compounds of cement.
Some of the most prominent examples are
given in Table 9-8 (Kriven, 1982, 1988).
Because of its technological interest as a
toughener for brittle ceramic materials, zir-
conia is considered as the prototype of
martensite in ceramic materials. On cool-
ing, the high-temperature cubic phase of
zirconia transforms at 2370 °C to a tetrago-
nal phase. On further cooling, bulk zirconia
transforms at 950 °C to a monoclinic phase
with a volume increase of 3%. The latter
transforms on heating at about 1170 °C.
The monoclinic to tetragonal phase trans-
formation is considered to be martensitic.
The M
stemperature can be lowered sub-
stantially even below room temperature by
alloying or by reducing the powder size.
Small particles of zirconia, embedded in a
single-crystal matrix of alumina, remain
metastable (= tetragonal) at room tempera-
ture for particle diameters less than a criti-
cal diameter (Rühle and Kriven, 1982).
These metastable particles can transform to
the monoclinic phase under the action of an
applied stress, and it is this property that is
exploited in toughening brittle ceramics
(see Becher and Rose (1994)).
Polymorphism is known to occur in
several crystalline polymeric materials. In
most of these systems the transformation
depends strongly on thermal activation.
However, in PTFE (polytetrafluoroethy-
lene) the conditions for no or weak thermal
activation are fulfilled, and the transforma-
tion can then be regarded as a diffusionless
640 9 Diffusionless Transformations
Figure 9-37.(a) Helix structure of the a- and b-
modification of PTFE, and (b) dilatometric deter-
mination of relative volume changes and tempera-
ture range of transformations of PTFE (Hornbogen,
1978).www.iran-mavad.com
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691. 9.10 Special Properties and Applications 641
or martensitic transformation (Hornbogen,
1978). This polymer crystallizes as parallel
arrangements of molecular chains parallel
to the c -axis. The atoms along the chains
are arranged as helices; the period along
the c-axis in the a-modification is 13C
2F
4
units while it is 15 units in the b-modifica-
tion. The transformation from the a- to the
b-helix occurs at about 19 °C (Fig. 9-37).
Relaxation of the helix during this transfor-
mation does not lead to an extension of the
specific length of the molecules in the c -di-
rection. The diameters of the molecule and
thus the lattice parameters in the a-direc-
tion increase, which leads to an increase
of about 1% in the specific volume. The
observed shape change can be increased if
the molecules have been aligned by plastic
deformation. An analysis of the shape
changes leads to the conclusion that the
PTFE transformation is diffusionless by a
free volume shear, a type of transformation
not yet known in metallic and inorganic
materials.
Biological materials consisting of crys-
talline proteins also undergo martensitic
transformations in performing their life
functions. In a review entitled “Martensite
and Life”, Olson and Hartman (1982) dis-
cuss some examples. The tail-sheath con-
traction in T4 bacteriophages can be de-
scribed as an irreversible strain-induced
martensitic transformation, while polymor-
phic transformations in bacterial flagellae
appear to be stress-assisted and exhibit a
shape-memory effect.
9.10 Special Properties
and Applications
9.10.1 Hardening of Steel
Much of the technological interest con-
cerns martensite in steels. In a review on
strengthening of metals and alloys, Wil-
liams and Thompson (1981) consider mar-
tensite as one of the most complex cases of
combined strengthening. The hardness of
martensite in as-quenched carbon steel de-
pends very much on the carbon content. Up
to about 0.4 wt.% C the hardening is lin-
ear; retained austenite is present in steel
containing more carbon, which reduces the
rate of hardening. Solute solution harden-
ing by the interstitial carbon atoms is very
substantial, whereas substitutional solid
solution hardening is low. For example,
Fe–30 wt.% Ni martensites, where the car-
bon content is very low, are not very hard.
The hardening of martensite is not due only
to interstitial solute solution hardening,
however. The martensite contains a large
number of boundaries and dislocations,
and the carbon atoms may rearrange during
the quench forming clusters that cause ex-
tra dislocation pinning. The various contri-
butions to the strength of a typical C-con-
taining martensite are given in Table 9-9,
from which it becomes evident that inter-
stitial solid solution hardening is not the
most important cause. Because many mar-
tensitic steels are used after tempering,
lower strengths than those shown in Table
9-9 are found.
9.10.2 The Shape-Memory Effect
A number of remarkable properties have
their origin in a martensitic phase trans-
Table 9-9.Contribution to as-quenched martensite
strength in 0.4 wt.% C steel (Williams and Thomp-
son, 1981).
Boundary strengthening 620 MPa
Dislocation density 270 MPa
Solid solution of carbon 400 MPa
Rearrangement of C in quench 750 MPa
Other effects 200 MPa
0.2% yield strength 2240 MPawww.iran-mavad.com
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692. formation, such as the shape-memory ef-
fect, superelasticity, rubber-like behavior
and pseudoelasticity. The most fascinating
property is undoubtedly the shape-memory
effect. Review articles were published by
Delaey et al. (1974), Otsuka and Shimizu
(1986), Schetky (1979), and Junakubo
(1987). Recently, Van Humbeeck (1997)
prepared a review on Shape Memory Mate-
rials: “State of the Art and Requirements
for Future Applications”. His review con-
tains 104 references to recent articles on
the topic.
A metallic sample made of a common
material (low-carbon steel, 70/30 brass,
aluminum, etc.) can be plastically de-
formed at room temperature. The macro-
scopic shape change resulting from this
deformation will remain unchanged if the
sample is heated to higher temperatures.
The only observable change in property
may be its hardness, provided that the tem-
perature to which the sample has been
heated is above the recrystallization tem-
perature. Its shape, however, remains as it
was after plastic deformation. If the sample
is made of a martensitic shape-memory
material and is plastically deformed (bent,
twisted, etc.) at any temperature below M
f
and subsequently heated to temperatures
above A
f, we observe that the shape that
the specimen had prior to the deformation
starts to recover as soon as the A
stempera-
ture is reached and that this restoration is
completed at A
f. This behavior is called the
“shape-memory effect”, abbreviated to SME.
If the SME sample is subsequently
cooled to a temperature below M
sand its
shape remains unchanged on cooling, we
talk about the “one-way shape-memory ef-
fect”. If it spontaneously deforms on cool-
ing to temperatures below M
sinto a shape
approaching the shape that it had after the
initial plastic deformation, the effect is
called the “two-way shape-memory ef-
fect”. A more visual description of these
two effects is given in Fig. 9-38 and a clar-
ifying example is shown in Fig. 9-39,
where the applicability of the one-way
shape-memory effect is given for a space-
craft antenna.
The shape that has to be remembered
must, first of course, be given to the speci-
men. This is done by classical plastic de-
formation by either cold or hot working.
This process, however, may not involve
any martensite formation. The material
must therefore be in a special metallurgical
condition, which may require additional
thermal treatments. In Fig. 9-40, for
example, depicting a temperature-actuated
shape-memory switch, two different “re-
membering” shapes are used. The initial
shape may be obtained by hot extrusion or
wire drawing and may or may not receive
an additional cold or hot working in order
to obtain the required shape. The shapes
formed must receive a heat treatment, con-
sisting of a high-temperature annealing,
followed by water quenching. The speci-
men is now martensitic, provided that the
composition is such that M
fis above room
temperature. In order to induce the shape
memory, the martensitic specimens are
bent either to be curved or to be straight
and are placed into the actuator at room
temperature. If the temperature of the actu-
ator exceeds the reverse transformation
642 9 Diffusionless Transformations
Figure 9-38.Schematic illustration of the shape-
memory effect: (a) and (e) parent phase; (b), (c) and
(d) martensite phase (Otsuka and Shimizu, 1986).www.iran-mavad.com
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693. 9.10 Special Properties and Applications 643
temperature of the shape-memory material,
the specimens recover towards their “re-
membered” position. The electrical con-
tacts are either closed or opened.
Special procedures for handling of the
shape-memory device are needed if we
want to induce the two-way memory effect.
This can be explained by again taking the
temperature-actuated switch as an exam-
ple. If the specimen taken in its remem-
bered position is cooled back to room tem-
perature, we do not expect further shape
changes to occur. In order to reuse the
specimens after having performed the
shape-memory effect, they must be bent to
be either curved or straight again. Reheat-
ing these deformed specimens for a second
time to temperatures above A
fwill result in
shape memory. If this cycle, bending–heat-
ing–cooling, is repeated several times,
gradually a two-way memory sets in. Dur-
ing cooling the specimen reverts spontane-
Figure 9-39.Application of
Nitinol for a shape-memory
spacecraft antenna. From
“Shape Memory Alloys” by
L. McDonald Shetky. Copy-
right (1979) by Scientific
American, Inc. All rights re-
served.www.iran-mavad.com
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694. ously to its “deformed” positions, thus
opening or closing the electrical contacts
on cooling. This repeated cycling, defor-
mation in martensitic condition followed
by a heating–cooling, is called “training”.
We can thus induce two-way memory by
using a training procedure.
A further comment should be made here
concerning the shapes that can be remem-
bered. We have to distinguish three shaping
procedures: the fabrication step from raw
material towards, for example, a coiled
wire such as for the antenna, the fabrica-
tion of the “to be remembered position”,
such as the additional shaping for the actu-
ator, and the final deformation in the mar-
tensitic condition, such as the bending of
the actuator. The first two fabrication steps
involve only classical plastic deformation
and, therefore, the type and degree of de-
formation are in principle not limited, pro-
vided that the material does not fail. The
degree of deformation, however, is limited
in the third deformation step, because it
may not exceed the maximum strain that
can be recovered by the phase transforma-
tion itself. Because these strains are asso-
ciated with the martensitic transformation,
the maximum amount of recoverable strain
is bound to the crystallography of the trans-
formation. Exceeding this amount of de-
formation in the third fabrication step will
automatically result in unrecoverable de-
formation.
Many examples of shapes that can be re-
membered are possible. A flat SME speci-
men can elongate or shorten during heat-
ing, can twist clockwise or counter-clock-
wise, and can bend upwards or downwards.
An SME spring can expand or contract dur-
ing heating. All this depends on the second
and third fabrication steps.
What happens now if, for one reason or
another, the specimen is restrained to ex-
hibit the shape-memory effect? For exam-
ple, what happens if an expanded ring is
644 9 Diffusionless Transformations
Figure 9-40.Temperature-actuated switch designed so that it opens or closes above a particular temperature.
From “Shape Memory Alloys” by L. McDonald Shetky. Copyright (1979) by Scientific American, Inc. All
rights reserved.www.iran-mavad.com
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695. 9.10 Special Properties and Applications 645
fitted as a sleeve over a tube with an outer
diameter slightly smaller than the inner di-
ameter of the expanded ring but larger than
the inner diameter of the ring in the remem-
bered position? On heating, the ring will
start to shrink as soon as the temperature A
s
is reached. While shrinking it will touch
the tube wall and further shrinking will be
hindered. From this moment, a compres-
sive stress will be built up, clamping the
shrinking ring around the tube. Obviously,
the composition of the alloy should be such
that the M
stemperature is below the value
at which clamping is required; in many
cases, this is below room temperature. For
clamping rings it is therefore important that
on cooling back to room temperature the
clamping stress is still present. This means
that two-way memory must be avoided,
which is easily achieved by choosing a
shape-memory alloy that exhibits a large
temperature hysteresis.
On heating a shape-memory device,
stresses can thus be built up and mechanical
work can be done. The latter would be the
case if a compressed shape-memory spring
has to lift a weight as in a shape-memory
actuated window opener (Fig. 9-41). A
very useful device is realized when a
shape-memory device is used, as shown
in Fig. 9-42, in combination with a bias
spring made of a conventional linear elastic
material, both being clamped between two
fixed walls and attached to each other with
a plate. At temperatures below M
f, the
shape-memory spring is closed and com-
pressed by the bias spring. The SME spring
had to be deformed, in this case com-
pressed, in order to fit into the clamping
unit. The clamping unit with the two
springs installed is now heated to tempera-
tures above A
f; as soon as A
sis reached, the
SME spring will start to expand and try to
push back the bias spring. At A
f, the shape-
memory spring will not yet have regained
its original length, and further heating is re-
quired to overcome the force exerted by the
bias spring. At a certain temperature higher
than A
fthe shape-memory spring will be
fully recovered. This temperature will de-
pend on the strength of the bias spring.
During this temperature excursion, the
plate that is fixed between the two springs
will have moved and can, if an “engine” is
attached to it, deliver work. If the clamping
unit is now cooled, the bias spring will try
to compress the shape-memory spring into
Figure 9-41.A simple shape-memory window
opener made from a copper-based shape-memory al-
loy. From “Shape Memory Alloys” by L. McDonald
Shetky. Copyright (1979) by Scientific American,
Inc. All rights reserved.
Figure 9-42.A mechanism in which a shape-mem-
ory alloy (SMA) spring is used in conjunction with a bias spring. From “Shape Memory Alloys” by L. McDonald Shetky. Copyright (1979) by Scientific American, Inc. All rights reserved.www.iran-mavad.com
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696. its deformed position. The elastic energy
that has been stored in the bias spring dur-
ing the heating cycle is now released, al-
lowing the plate to perform work also dur-
ing the cooling cycle. To describe fully
such a working performing cycle, a ther-
modynamic treatment is needed (Wollants
et al., 1979). The working performing
cycle can best be illustrated by taking a
shape-memory spring that expands or con-
tracts during heating or cooling and that
carries a load. The working performing cy-
cle can then be represented in a displace-
ment–temperature, a stress–temperature,
or an entropy–temperature diagram.
Although the shape-memory effect has
been observed in many alloy systems, only
three systems are commercially available,
mainly because of economic factors and
the reliability of the material. The three al-
loy systems are Ni–Ti, Cu–Zn–Al and
Cu–Al–Ni. Generally, other elements are
added in small amounts (of the order of a
few weight %) in order to modify the trans-
formation temperatures or to improve the
mechanical properties or the phase stabil-
ity. In all three cases the martensite is
thermoelastic. Maki and Tamura (1987)
reviewed the shape-memory effect in fer-
rous alloys, where a non-thermoelastic
Fe–Mn–Si alloy has also been found to
show a shape-memory effect, and commer-
cialization is being considered. The most
important properties of shape-memory al-
loys are summarized in Fig. 9-43, in which
the working temperatures, the width of the
646 9 Diffusionless Transformations
Figure 9-43.Schematic representation of the most relevant shape-memory properties (courtesy Van Hum-
beeck, 1989).www.iran-mavad.com
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697. 9.10 Special Properties and Applications 647
hysteresis, and the maximum recoverable
strain are given. Because of the superior
mechanical, chemical and shape-memory
properties of Ni–Ti alloys, this alloy sys-
tem has been applied most successfully;
about 90% of the present applications use
these alloys. Owing to the continuous im-
provement of the properties of Cu-based
alloys, together with their lower price, Cu
SME alloys have been successfully used in
several applications.
The commercial applications of shape-
memory devices can be divided into four
groups:
1. motion: by free recovery during heat-
ing and/or cooling;
2. stress: by constrained recovery during
heating and/or cooling;
3. work: by displacing a force, e.g., in
actuators;
4. energy storage: by pseudoelastic load-
ing of the specimen.
Shape-memory effects have also been
reported in non-martensitic system, e.g.,
in ferroelectric ceramics (Kimura et al.,
1981), and have found applications as
micro-positioning elements (Lemons and
Coldren, 1978). The shape change is attrib-
uted here to domain-wall motion, as shown
in Fig. 9-44.
9.10.3 High Damping Capacity
The hysteresis exhibited during a pseu-
doelastic loading and unloading cycle is a
measure of the damping capacity of a vi-
brating device fabricated from a shape-
memory material, which is cycling under
extreme stress conditions exceeding the
critical stress needed to induce martensite
by stress. Vibrating fully martensitic sam-
ples also exhibit high damping. A fully
martensitic sample consists of a large
number of differently crystallographically
oriented domains whose domain boundar-
ies are mobile. Under the action of an ap-
plied stress these boundaries move but, be-
cause of friction, energy is lost during this
movement. If a cyclic stress is applied, this
foreward and backward boundary move-
ment will lead to damping of the vibration.
Comparing the amount of this damping
with the damping that we observe in other
non-SME alloy systems, it is found that the
damping capacity of martensitic shape-
memory alloys is one of the highest. The
shape-memory alloys are said to belong to
the high-damping materials, the so-called
HIDAMETS.
9.10.4 TRIP Effect
TRIP is the acronym for TRansforma-
tion-Induced Plasticity and occurs in some
high-strength metastable austenitic steels
exhibiting enhanced uniform ductility
when plastically deformed. This uniform
macroscopic strain, up to 100% elongation,
accompanies the deformation-induced
martensitic transformation and arises from
a plastic accommodation process around
the martensite plates. This macroscopic
strain thus contrasts with that occurring in
Figure 9-44.Schematic illustration of the mecha-
nism for an electronic micro-positioning (Lemons
and Coldren, 1978).www.iran-mavad.com
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698. shape-memory alloys in being unrecover-
able.
TRIP has been extensively studied by
Olson and Cohen (1982a) and we will fol-
low their approach here. They distinguish
two modes of deformation-induced trans-
formation, according to the origin of the
nucleation sites for the martensite plates:
“stress-assisted” and “strain-induced” trans-
formation. The condition under which each
mode can operate is indicated in a tempera-
ture–stress diagram as shown in Fig. 9-45.
At temperatures slightly higher than M
s
s
,
the stress required for stress-assisted nucle-
ation on the same nucleation sites follows
the line AB. At B, the yield point for slip
in the parent phase is reached, defining
the highest temperature M
s
s
for which the
transformation can be induced solely by
elastic stresses. Above this temperature,
plastic flow occurs before martensite can
be induced by stress. New strain-induced
nucleation sites are formed, contributing to
the kinetics of the transformation. The
stresses at which this strain-induced mar-
tensite is first detected follows the curve
BD. At point D, fracture occurs and thus
determines the highest temperature M
dat
which martensite can be mechanically in-
duced.
When the transformation occurs at tem-
peratures below M
s
s
, the plastic strain is
due entirely to transformation plasticity re-
sulting from the formation of preferential
martensite variants. The volume of the in-
duced martensite is therefore linearly re-
lated to the strain. The existing nucleation
sites are aided mechanically by the thermo-
dynamic contribution of the applied stress,
reducing the chemical driving force for
nucleation. Above M
s
s
, the relationship
between strain and volume of martensite
becomes more complex, because strain is
648 9 Diffusionless Transformations
Figure 9-46.Transformation-induced plasticity in
tensile tests at various temperatures (Fe–29 wt.%
Ni–0.26 wt.% C) (Tamura et al., 1969).
Figure 9-45.Idealized stress-assisted and strain-in-
duced regimes for mechanically-induced nucleation (Olson and Cohen, 1982a).www.iran-mavad.com
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699. 9.11 Recent Progress in the Understanding of Martensitic Transformations 649
then a result of plastic deformation of the
parent phase and of transformation plastic-
ity. Strain hardening and enhancement of
nucleation of martensite also play an essen-
tial role. When martensite is formed during
tensile deformation, the strain hardening
becomes large. Necking is then expected
to be suppressed, explaining the enhanced
uniform elongation. Fig. 9-46 shows, as an
example, the amount of martensite, the
elongation and the ultimate strength meas-
ured after tensile tests of a TRIP steel as a
function of temperature, clearly illustrating
the enhanced elongation, especially in the
temperature range between M
s
s
and M
d.
Such a large elongation (sometimes over
200%) can also be produced by subjecting
a TRIP steel specimen under constant load
to thermal cycles through the transforma-
tion temperature.
9.11 Recent Progress in the
Understanding of Martensitic
Transformations
We draw attention here to some recent
papers that illustrate recent progress in the
understanding of martensite, and in which
some new approaches are also explained.
Most of the information referred to in this
section was presented at the most recent
ICOMAT international conference on mar-
tensitic transformations held in 1998 at
Bariloche (Ahlers et al., 1999).
New directions in martensite theory are
presented by Olson (1999). The nucleation
of martensite, the growth of a single mar-
tensite plate, the formation of, for example,
self-accommodating groups of martensite
plates, and this within single crystals of the
parent phase as well as in polycrystalline
material, and the constraints dictated by the
components where martensitic materials
are only one (maybe the most important)
functional element of the component, are
all influenced by different interactive lev-
els of structures (ranging from solute atoms
to components). Nucleation is the first step
in martensite life, and a component whose
functional properties are attributed to those
of martensite, can be considered the final
step. Olson (1999) constructed a flow-
block diagram in which the martensitic
transformation is situated in a multilevel
dynamic system. This new system, shown
in Fig. 9-47, and the one given in Fig. 9-1,
offer powerful tools for a better under-
Figure 9-47.The flow-block diagram
of martensitic transformation as a
multilevel dynamic system (Olson,
1999).www.iran-mavad.com
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700. standing of diffusionless phase transforma-
tions. Its use should lead to a better design
of martensitic and bainitic alloys meeting
specific requirements. Close analysis of the
papers presented at ICOMAT 98 shows that
such an approach can already be found in
many papers.
Special attention is given to the influ-
ence of external constraints, such as hydro-
static pressure, the application of a mag-
netic field, and to martensite formed in thin
films prepared by either sputtering or rapid
solidification. Kakeshita et al. (1999) stud-
ied the influence of hydrostatic pressure in-
stead of uniaxial stress, in order to formu-
late a thermodynamic approach for a better
understanding of the nucleation of marten-
site. The strengthening mechanisms in
steel due to martensite are reviewed, the
diffusion of carbon in the various states
(according to the dynamic system of Fig.
9-47) of martensite is highlighted. In this
context, the fracture mechanism is related
to the tempering temperature and the car-
bon diffusion.
As is commonly known, the mechanism
of bainite transformation is a subject with
many unresolved issues. Bhadeshia (1999)
gives an overview of the transformation
mechanisms proposed to explain “among
others” the growth of bainite. The develop-
ment of bainite at both high temperatures
(upper bainite) and low temperatures
(lower bainite) is discussed and is illustrat-
ed in Fig. 9-48. According to Bhadeshia,
the unresolved issues are:
the growth rate of an individual bainite
plate
a theory explaining the kinetics to esti-
mate the volume fraction of bainite in
austenite obtained during an isothermal
transformation
the modeling describing quantitatively
the formation of carbides
and a number of features associated with
the interaction between plastic deforma-
tion and bainite formation.
The influence of carbon on the bainitic
transformation is treated in great detail and
is shown to be a controlling factor of the
mechanical properties of different multi-
phase TRIP-assisted steels (Girault et al.,
1999; Jacques et al., 1999).
The martensitic transformation in
Fe–Mn-based alloys is treated in various
papers, showing the increasing interest in
developing ferrous shape-memory alloys.
In these alloys, austenite transforms either
into a h.c.p. e-phase (g
¨Æe) or/and into
a¢-martensite (g
¨Æa¢).
New approaches and strategies are dis-
cussed for the application of shape-mem-
ory alloys in non-medical (Van Humbeeck,
1999) as well as in medical applications
(Duerig et al., 1999). Only two examples
are shown here. The first example (Fig.
9-49) shows that Ni–Ti superelastic alloys
improve significantly the cavitation ero-
sion resistance if compared with marten-
650 9 Diffusionless Transformations
Figure 9-48.A schematic representation of the
mechanism explaining the growth and development
of bainite (Bhadeshia, 1999).www.iran-mavad.com
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701. 9.12 Acknowledgements 651
sitic Ni–Ti. But it should be remarked that
this figure is only an enlargement of a fig-
ure giving an overall view of the cavitation
resistance of other common alloys. For ex-
ample, the weight loss after 10 h is already
20–30 mg for the “common” alloys in
comparison with the negligible weight loss
of both Ni–Ti shape-memory alloys after
10 h tested under the same conditions. A
very impressive example of the application
of Ni–Ti shape-memory alloys is given in
Fig. 9-50. This figure shows an atrial septal
occlusion device with Nitinol (Ni–Ti
shape-memory alloy) wires incorporated in
a sheet of polyurethane. This device allows
holes in the atrial wall of the heart to be
closed without surgery. The two umbrella-
like devices are folded in two catheters,
which are placed on either side of the
hole. Once the two folded umbrellas are
withdrawn from their catheters, they are
screwed together in such a way that the
hole is closed. Because of the flexibility of
both materials, the heart can again beat
normally. This device illustrates the con-
cept of the elastic development capacity
of shape-memory alloys. Because Ni–Ti
shape-memory alloys have proposed to be
biocompatible (see Van Humbeeck, 1977),
many applications of these Ni–Ti alloys
are currently being developed and mar-
keted.
9.12 Acknowledgements
The author would like to thank M. Ah-
lers, J. W. Christian, M. De Graef, R. Gott-
hardt, P. Haasen, H. S. Hsu, J. Ortín, K. Ot-
suka, J. Van Humbeeck and P. Wollants for
support and advice while preparing the
manuscript, and M. Van Eylen, M. Nol-
mans, H. Schmidt and K. Delaey for their
assistance. The “Nationaal Fonds voor We-
tenschappelijk Onderzoek” of Belgium is
acknowledged for financial support (pro-
ject No. 2.00.86.87). The author especially
acknowledges the continuing interest and
encouragement he received from A. De-
ruyttere. For help in preparing the revised
version, I would like to thank M. Chandra-
sekaran.
Figure 9-49.The weight loss of a martensitic (Ni
Ti–1) and a pseudoelastic (Ni Ti–2) Ni–Ti shape-
memory alloy (Richman et al., 1994).
Figure 9-50.A shape-memory device for repairing
defects in the heart wall (Duerig et al., 1999).www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

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654 9 Diffusionless Transformationswww.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

705. Index
Aaronson–Aaron model 506
absolute stability limit, directional dendritic
growth 148
acetone, dendritic growth 115
acoustic phonons 258
activation energy, diffusion 201, 491
activity coefficient
– decomposition 351
– Henrian 30
– phase diagrams 14, 70
adjoining phase regions law 47 f, 56 f
Ag–Au system 564
Ag-based alloys 598 f
Ag–Cd system 632
Ag–Cu system 150
Ag–Mg system 30
age hardening alloys, microstructure 401 ff
aggregation
– clusters 469
– diffusion 191
aging parameters 314 ff, 321 ff, 325 ff
Akaiwa–Voorhees model 373
Al–Ag–Zn system 398
Al–Ag system 488
Al–Cu system
– directional dendritic growth 150
– precipitation 325
Al–CuAl
2
alloys 161
Al–Fe system 150
Al–Li alloys, coarsening 375
Al–Mg–Si alloys 325
Al–Ni–Co system 377
Al–Zn–Ag system 329
Al–Zn–Mg system
– phase diagram 45
– precipitation 397
Al–Zn system
– dynamic fluctuations 277
– interfacial diffusion 510
– precipitation 397, 505, 510
– spinodal decomposition 451 ff
alkali alloys 634
alkali halides 639
Allen–Cahn concept 486 f
allotropic transformations
– interfacial diffusion 499
– martensites 634
alloys
– atomic ordering 523 ff
– dendritic growth 119
– diffusion 199, 221 ff
– diffusionless transformations 583 ff
– eutectic growth 158
– order–disorder phenomena 283 ff
– precipitation 314 ff
– spinodal decomposition 451
– spinodal 363 ff
amorphization, pressure-induced 661
amorphous alloys 366, 454
amplification factor, decomposition 357
analytical electron microscopy (AEM) 327
analytical field ion microscopy (AFIM) 328, 360, 366
angle dispersive X-ray powder diffraction
(ADX) 678, 680
anharmonic terms, phase transitions 292
anion–anion distances, high-pressure
transformations 663
anisotropic attachment kinetics, dendritic growth 115
anisotropic interactions, atomic ordering 558
anisotropy, phase transitions 268, 280
annealed defects 468
annealed disorder 279
annealing, shape memory 643
antiferromagnets 273, 297
antiphase boundaries 487
anvils 670 ff
Arrhenius behavior, diffusion 191, 199 ff, 216 ff
associative jumps, vacancies 197
athermal transformations 583–654
atom–vacancy exchange 218
atomic jumps 189
atomic ordering 519–581
attachment kinetics, anisotropic 115
Au–Cd alloys 590
Au–Cu system, atomic ordering 561, 564
Au–Ni system
– atomic ordering 563 f
– phase diagram 21
Au-based alloys, martensite 599 f
austenite 588, 600, 617
austenite stabilizing, martensitic transformations 625
autocatalytic growth, martensite 618 f
Phase Transformations in Materials. Edited by Gernot Kostorz
Copyright © 2001 WILEY-VCH Verlag GmbH, Weinheim
ISBN: 3-527-30256-5www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

706. 698 Index
average jump frequencies 199
averaging, phase transitions 279
Avrami equation 401, 465
B–C–N system, high pressure 683 ff
B–C system, high pressure 686
B–N system, high pressure 685
b.c.c. structures 537, 558 f, 577
bacterial flagellae, martensitic transformation 640
Bain strain 594 f, 601 f, 620
bainite 650
band–dendrites transitions 150
banded structure, diffusionless transformations 611 f
Be–Cu alloys 677
Becker–Döring theory
– decomposition 342 ff, 352 f
– precipitation 320 ff, 395
belite, diffusionless transformations 639
belt apparatus, high-pressure transformations 669
bending–heating–cooling cycles 644
beta alumina, diffusion 209
Bethe approximation
– atomic ordering 524
– phase transitions 287
Bi–Sn–Cd system 40, 44 ff
bicritical points, phase transitions 275
bicrystals
– discontinuous precipitation 506
– interfacial diffusion 490
bifurcation
– directional solidification 127 f
– precipitation 337
bilinear coupling 280
binary alloys
– diffusion 200
– phase transitions 285, 296
– point variables 527
– precipitation 314, 317 f, 397
– substitutional, diffusion 226
binary fluids, spinodal decomposition 455
binary mixtures
– Ising model 443
– spinodal decomposition 414 ff
binary monotectic reaction 26
binary systems
– atomic ordering 523 ff
– diffusion 208 f
– eutectic 23
– liquid 246
– metallic 503
– phase diagrams 14 ff, 42, 314
Binder modelsee:cluster dynamics/model
biological materials, martensitic transformations 640
Bitter–Crum theorem 448
bivariant phase regions 31
block copolymer melts 259, 271
body-centered cubicsee:b.c.c.
Boltzmann–Matano analysis 177 ff, 223
bond model, diffusion 200, 227
bonding energy, phase diagrams 12
Born approximation, electrical resistivity 469
boron nitride, displacive transformation 639
boundary conditions
– dendritic growth 102 f, 108
– diffusion 176 ff
– interfacial diffusion 484 ff, 503
– LBM theory 423
– solidification 91 ff, 122
boundary motion 495 f
Bragg–Williams approximation 255, 287
Bragg peak
– ordering 463 ff
– phase transitions 250, 258, 293
– solidification 87
Bridgman solidification 96 f, 668 ff
Brillouin scattering 675
Brown’s bond valence approach 664
Buerger deformations 611
bursts, martensitic transformations 633, 636
C–N system, high pressure 686
Cahn–Hilliard–Cook approximation
358 f, 420 ff,438 ff, 448
Cahn–Hilliard model
– decomposition 346 ff
– interfacial diffusion 486 ff
see also:continuum model
Cahn treatment
– interface migration 496 ff, 507
– spinodal decomposition 437
calcite–argonite transition 259
calculation models, ternary from binary data 72 ff
see also:Monte Carlo
CALPHAD method
– coarsening 373
– concomitant processes 394
– phase separation 323 ff
CaO–Fe
2
O
3
system, phase diagram 53
CaO–MnO system, phase diagram 14
CaO–SiO
2
system, phase diagram 27
capillarity
– decomposition 340
– directional solidification 124
capillary-corrected diffusion 158
capillary forces, interface migration 492
capillary length, dendritic growth 108, 125, 131
carbides
– atomic ordering 523
– diffusionless transformations 650
carbon
– diffusionless transformations 639
– high-pressure transformations 684
carbon nitrides, high pressure 686
CASCADE computer code 219
cast duplex steels, phase separation 366
casting
– eutectic growth 158
– solidification 96, 120
catastrophic nucleation 344
catatectic invariants 32www.iran-mavad.com
+ s e l '4 , kp e r i &s ! 9 j+ N 0 e

707. Index 699
cation diffusion 231
Cd–Al–Mn system 562
Cd–Mg system 563, 573
Cd–Na system 36 f
cell spacing, directional solidification 130 ff
cellular growth
– dendrites 140 ff, 146
– solidification 85, 126 f
cements, displacive transformation 639
ceramic solutions, phase diagrams 64
ceramics
– diffusionless transformations 639
– spinodal decomposition 454 f
chain polymers, diffusionless transformations 639
charge carrier distribution, high-pressure
transformations 666
charge diffusion coefficient 212
chemical composition gradient 178
chemical diffusion 184 ff, 232
– binary system 209
– dendritic growth 108
– directional solidification 124
chemical equilibrum 6
chemical forces
– interface migration 493, 503
– phase separation 322
chemical interdiffusion 178 f
chemical potential 10
chemically induced grain boundary migration
(CIGM) 503 ff
Chemla experiment 181
chill plate directional solidification 98
classification, displacive transformations 587 f, 634
Clausius–Clapeyron equation
– diffusionless transformations 607
– directional solidification 126
– martensitic transformations 629
cluster–cluster aggregation 469
cluster–diffusion–coagulation mechanism
– coarsening 377, 399
– spinodal decomposition 452, 456 f
cluster correlation functions 529 ff
cluster dynamics approach
– concomitant processes 394 f
– nucleation 352
– spinodal decomposition 432, 472
cluster interactions, atomic ordering 548
cluster kinetics 340 f
cluster probabilities
– atomic ordering 529 f
– phase transitions 289
cluster site approximation (CSA) 560
cluster variation method (CVM)
– atomic ordering 551, 556 f, 573
– decomposition 345
– phase transitions 288 f
clusters
– atomic ordering 524 ff, 529 ff
– concomitant processes 381, 391
– decomposition 340 ff
– ionic crystals 230
– precipitation 315 ff
– spinodal decomposition 415 ff, 444
Co–Al system, ordering 570
Co–Ni–O system 51
coagulation
– clusters 352
– coarsening 377
– spinodal decomposition 433, 446 f, 452
coarse graining
– decomposition 347
– phase transitions 255, 263, 300
– spinodal decomposition 414 f, 418 ff, 438 ff
coarsening
– concomitant processes 381 ff
– decomposition 326, 365
– precipitation 315, 321 ff, 336, 370 ff
– spinodal decomposition 416, 445
coarsening stages, experimental identification 387
cobalt alloys, diffusionless transformations
608, 617, 634 ff
coexistence curve 414
coherency strains 377
coherent elastic misfit 447
coherent interfacial energies 390
coherent miscibility gaps 571
coherent phases, precipitation 324
coherent potential approximation (CPA), effective
cluster interactions 550
coherent solvus lines 316
collective correlation factors, diffusion 206
collective diffusion coefficients 179
colloid crystallization 466
columnar growth 95
composition triangle, ternary 39
compositions
– atomic ordering 544, 566
– eutectic 23
– phase transitions 246
see also:stochiometric compositions
compounds
– phase diagrams 9, 65
– stoichiometric 29
compressibility 662
computer simulation techniques, phase transitions
283, 293
concentrated alloy systems 199, 226
concentration–depth profile 231
concentration fluctuations 339
concentration gradients 15, 175
concentration profile
– diffusion coefficients 235
– interfacial diffusion 502
concentration waves
– interfacial diffusion 486
– spinodal decomposition 415 f, 434
concomitant nucleation, spinodal decomposition 433
concomitant processes, precipitation 381 ff, 401
configuration polyhedron, atomic ordering 531 ff
configurations, atomic 523 ff
congruent compounds 29
conjugate phases 336www.iran-mavad.com
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708. 700 Index
conjugate thermodynamic variables 60, 247
conode 15
conservation laws, phase transitions 277
consolute temperature 21
constant composition sections, phase diagrams 46
constrained growth 95
continuity equation, solidification 92 ff
continuous casting 96
continuous cooling 501
continuum model
– decomposition 346
– diffusion 175
– interfacial diffusion 486 f
– spinodal decomposition 356 f, 421
conventional transmission electron microscopy
(CTEM), decomposition 326, 393
cooling
– diffusionless transformations 587
– interfacial diffusion 501
– phase diagrams 15
– precipitation 314 f, 401
– shape memory effect 644
coordination number
– atomic ordering 524, 539
– high-pressure transformations 663
copper, alloying elements, diffusionless
transformations 611, 634
correlation factor, diffusion 178, 193
correlation functions
– atomic ordering 526 ff, 567
– phase transitions 262
– solidification 87
– spinodal decomposition 430
correlation length
– intrinsic 85
– phase transitions 249
corresponding pairs, phase diagrams 60
Cottrell solute drag 496
Coulomb potentials
– colloidal suspensions 24
– pair interactions 294
coumarin-succinonitrile, solidification 141
coupling mechanisms, defects–order parameter
280
Cowley–Warren short-range order parameters 531
cristobalite, diffusionless transformation 639
critical amplitude, spinodal decomposition 438
critical clusters 434
critical exponents 260 f
critical nucleus 341
critical phenomena, phase transitions 260 f
critical radius, nucleation 88, 381, 392
critical temperature
– phase diagrams 21
– spinodal decomposition 416
cross-boundary diffusion 497 f
crystal growth, solidification 91
crystallization
– colloid 466
– primary 41
crystallographic relations 593 f, 614 f, 620
Cu–Ag–Au alloys, diffusionless transformations
607
Cu–Ag system, diffusionless transformations 564
Cu–Al–Mn system, diffusionless transformations 562
Cu–Al–Ni system
– diffusionless transformations 602, 621 f
– shape memory effect 645
Cu-based alloys, diffusionless transformations 598 f
Cu–Bi–Fe system, decomposition 397
Cu–Cd alloys, discontinuous precipitation 513
Cu–Co system
– decomposition 345, 350
– discontinuous precipitation 505
Cu–Mn system, phase separation 397
Cu–Ni–Fe system, decomposition 329, 334, 365
Cu–S–O system 7
Cu–Sn system, martensitic transformations 617, 622
Cu–Ti system
– decomposition 317, 325, 331, 350, 388
– interfacial energies 390, 512
Cu–Zn–Al system
– diffusionless transformations 598 ff
– shape memory effect 645
Cu–Zn system, martensitic transformations 617
Curie–Weiss law, phase transitions 261
cycle procedures, shape memory effect 644
cycling behavior, martensitic transformations 633
cyclohexanol, dendrites 117
damping capacity, diffusionless transformations
593, 636, 647
Darken–Manning relation 185 ff, 209
Datye–Langer theory 155, 157
Dauphiné twins 601, 606
decomposition
– experimental techniques 326 ff
– precipitation 309–408
– spinodal 409–480
defects
– diffusion 193, 201, 213
– eutectic growth 161
– interfacial diffusion 484 f
– ionic crystals 227, 231
– phase transitions 280 f
– precipitation 401
– spinodal decomposition 415, 468
deformation, diffusionless transformations 592, 615
degeneracy, solidification 92
dendrites 97
dendritic growth 85 ff, 100 ff
density differences, phase transitions 246
deviatoric components, diffusionless
transformations 588
DEVIL computer code 219
devil's staircase 274
Devonshire–Ginzburg–Landau theory 603 f
diamond, high-pressure transformations 683
diamond anvil cells (DAC) 668 f, 671, 678
diblock copolymers, phase separation 272
dielectric solid systems, phase transitions 246www.iran-mavad.com
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709. Index 701
differential scanning calorimetry, diffusionless
transformations 590
diffraction techniques
– decomposition 334
– diffusionless transformations 587, 611
– high-pressure transformations 671, 678 ff
diffuse interface model 347 f
diffusion–coagulation mechanism 446
diffusion
– coarsening 377
– dendritic growth 108
– directional solidification 124
– in crystals 171–238
– interfacial 481–518
– solidification 91
diffusion constants
– definitions 175 ff
– effective, short circuits 187
– spinodal decomposition 422
diffusion-controlled growth
– precipitation 350, 370 f, 374, 381
– solidification 94
– supersaturated matrix 350
diffusion couples 568
diffusion kinetics
– Cahn–Hilliard 420 ff
– solids 171–238
diffusion length
– directional solidification 126
– eutectic growth 153
diffusion-limited aggregation (DLA) 101
diffusionless transformations 583–654
diffusivity 175, 485 f
dilatation dominant transformations 587 f, 610
dilute alloys, diffusion 221, 226
dimensionality
– critical 268
– order parameters 250
direct imaging techniques 326 f
directional dendritic growth 131 ff
directional solidification 86, 95, 98, 120 ff
discontinuous precipitation 484 f, 504 f
discrete lattice point model 346 f
dislocations
– diffusion 186
– diffusionless transformations 620
– interfacial diffusion 485, 495
– spinodal decomposition 415
displacement temperature, shape memory effect 645
displacive transformations 292, 587–651
disruptive phase transitions 259
dissipation, spinodal decomposition 420
dissociative mechanism, diffusion 191, 197
distortions
– high-pressure transformations 659
– phase transitions 283
divacancies, diffusion 219
domain structures
– diffusionless transformations 600
– phase transitions 259, 274
doublons, solidification 114, 144
drift terms, diffusion 175, 183
driving forces
– interface migration 492
– martensitic transformations 624
– phase separation 322
droplets
– decomposition 340
– phase transitions 300
– spinodal decomposition 415, 418, 442 f
Dufour effect 181
dumbell split configuration, diffusion 190
duplex stainless steel, phase separation 364
dynamic pressure generation 667
dynamic scaling
– phase transitions 277 f
– precipitation 395 ff
see also:scaling
dynamics, phase changes 414 f
E–pH diagrams 9
early stage decomposition
– concomitant processes 381 f
– kinetics 339 ff
Eckhaus band instability, solidification 128 f, 135 f
effective cluster interactions (ECIs) 548
effective diffusion coefficients 187
Einstein equation 185, 192
elastic constants
– diffusionless transformations 608
– precipitation 336
elastic interaction, decomposition 447 f
elastic phase transitions 257
elastic strain
– diffusionless transformations 606 f, 636
– interface migration 493
– precipitation 324
– spinodal decomposition 447 f
electric fields, phase transitions 246
electrical resistivity
– diffusionless transformations 587
– spinodal decomposition 468
electron back-scattered diffraction (EBSD),
solidification 144
electron energy loss spectroscopy (EELS), phase
separation 327
electronic defects, ionic crystals 231
electronic properties, high-pressure
transformations 659
Ellingham diagrams 8 f
energy dispersive X-ray diffraction (EDX)
– decomposition 327
– high-pressure transformations 680
enstatite, diffusionless transformations 639
enthalpy
– diffusion 217
– diffusionless transformations 636
– phase diagrams 5, 11, 34
– precipitation 318
entropy
– diffusionless transformations 607, 636www.iran-mavad.com
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710. 702 Index
– directional dendritic growth 149
– directional solidification 121
– high-pressure transformations 660
– phase diagrams 5, 11, 34
– phase transitions 247
entropy–temperature diagram, shape memory effect
645
equiaxed growth, solidification 95
equilibrum cooling 15
equilibrum properties, solid–solid interfaces 484 ff
Eshelby theory, displacive transformations 606
Euler–Lagrange equation, second order transitions
261
eutectic growth 86, 99, 151 ff
eutectic invariants 31 f, 41
eutectic systems 22
eutectic temperature 23
eutectoid invariants 32
eutectoid reactions, interfacial diffusion 505
eutectoids 159
eutecular invariants 52
evaporation–condensation mechanism, spinodal
decomposition 456
evaporation, phase transitions 246
excess properties
– phase diagrams 13 ff
– polynomial representation 34
exchange mechanism
– atomic ordering 539, 569
– diffusion 191, 217 f
– phase transitions 284
existence domains, atomic ordering 531 ff
extended X-ray absorption fine structure
(EXAFS)
– atomic ordering 563
– high-pressure transformations 676
extension rules, phase diagrams 32
extrinsic defects, diffusion 230
f.c.c–b.c.c. transformations 594
f.c.c. structures, atomic ordering 532 ff, 540 ff, 555 f
faceted systems, eutectic growth 158
faceting transitions 489
FACT (Facility for the Analysis of Chemical
Thermodynamics), expert system 75
fast growth rates, solidification 98
fast-mode theory, spinodal decomposition 428
faulting, diffusionless transformations 622
faults, eutectic growth 161
Fe–Al system
– atomic ordering 562, 565
– order parameters 250
– phase diagram 276
Fe–C–Ni system, martensitic transformations 617
Fe–C system, martensitic transformations 617
Fe–Co–Al system, martensitic transformations
562, 569 f
Fe–Cr–Co system
– coarsening 377
– decomposition 329, 332
Fe–Cr–Ni system, martensitic transformations 617
Fe–Cr–O system, phase diagram 52
Fe–Cr–V–C system, phase diagram 48
Fe–Cr system, phase separation 397
Fe–Cu system, phase separation 350, 390
Fe–Mn–Si system, shape memory effect 644
Fe–Mn system
– martensitic transformation 617
– shape memory effect 650
Fe–Ni system, martensite 590, 617
Fe–Ni–C alloys 619
Fe–Ni–Cr system, precipitation 364
Fe–Pd alloys, diffusionless transformation 608, 613
Fe–Pt alloys, diffusionless transformation 607
Fe–Ru system, martensitic transformation 627
Fe–S–O system, phase diagram 8
Fe–Ti–Al system, ordering 562, 569
feldspar frameworks, high pressure 665
ferroelectric transitions 246
ferroic transformations 608
ferromagnetic systems, atomic ordering 570
ferrous martensite 366
ferrous systems, diffusionless transformation
607, 635
fibers, solidification 152
fibrous systems, eutectic growth 158
Fick's laws of diffusion 175–235
field ion microscopy (FIM), phase separation 326
finite precipitate volume fractions 373
first neighbor interactions 532, 540, 555 f
first-order phase transitions 249, 271
– martensites 599
– solidification 86 ff
Fisher renormalization 269, 279
five-frequency model, impurity diffusion
197, 207, 215
flat interfaces, crystal growth 91
Flory–Huggins approximation 426 f
fluctuations
– dendritic growth 113
– heterophases 302
– phase transitions 261, 271, 277
– precipitation 319
– solidification 87
– spinodal decomposition 414 f
fluid–suprafluid transition 246
fluid magnet analogy, phase transformations 247
fluids, spinodal decomposition 437 f, 455 f
fluorescence line, high pressure 672
flux balance, eutectic growth 153
force–velocity relationship, interface migration
492, 496
formation energies, phase diagrams 29
fractal structure 266
framework flexion, high-pressure
transformations 661
free growth
– dendritic 100 ff, 118
– solidification 96
freezing 91 ff, 120
Frenkel defects 213, 220, 228www.iran-mavad.com
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711. Index 703
frictional forces, interface migration 496
frozen-in clusters 444
frozen-in disorder 468
frozen-in impurities 279
frozen phonons 259
Furukawa function 447, 453
fusion temperature, phase diagrams 15
gallium arsenide, high pressure 681
gaseous compounds, phase diagrams 9
gasketing system, high pressure 670
Gaussian noise, phase separation 420
Ge–Si system, phase diagram 17
gel–gel transitions 456
germanium, high pressure 680
Gibbs–Duhem equation 11 ff, 30, 61
Gibbs–Thomson equation
– decomposition 351, 370 f, 382 ff, 394
– eutectic growth 154
– interfacial diffusion 492, 510
Gibbs absorption isotherm 490
Gibbs energy 5 ff
– minimization 73, 6 f
Gibbs phase rule 7 f
Ginzburg–Landau equation
– diffusionless transformations 603
– interfacial diffusion 487
– solidification 89
Ginzburg criterion
– phase transitions 263 ff, 300 ff
– spinodal decomposition 423, 431, 462, 471
girdle-type press 669
glasses, phase separation 361, 367, 454 f
glissile interfaces, martensitic transformations 622
gold alloys, displacive transformations 635
Gorsky effect, diffusion 233
grain boundaries
– diffusion 186
– migration 503 ff
– phase transitions 282
– precipitation 317, 325
– spinodal decomposition 415
grain refinement 119
graphical integration analysis, diffusion 233
graphite, transformations 6, 683
gravity effects, spinodal decomposition 456
Greninger–Troiano relations 616, 637 f
ground states, atomic ordering 538 ff
group formation, diffusionless transformations 619
group theoretical techniques 255
growth
– bainite 650
– decomposition kinetics 332
– dendritic 116, 148
– diffusion-controlled 350
– diffusionless transformations 636
– interface migration 495
– martensites 599, 632
– precipitation 319 ff, 370
– solidification 93 ff
– spinodal decomposition 415
Gruzin method, diffusion 232
Guggenheim model 68
Guinier–Preston zones 325, 360
Guinier approximation, small-angle scattering 391
gyration radius 426
habit plane 595 ff, 615 f, 622, 637
halides, diffusionless transformations 639
Hamiltonians, phase transitions 247 ff, 255 ff, 262 f,
271, 281 ff, 419
hardening
– diffusionless transformations 648
– steel 641
Hart equation 187
Haven ratio 183 ff, 190, 196, 211 f
heat flow, solidification 181
heat sinks, solidification 120
heating–cooling cycles, shape memory effect 644
heating, diffusionless transformations 590, 607
heating rate, decomposition 330
Heisenberg antiferromagnet 297
Heisenberg model 267
Hele–Shaw flow 106
helimagnetitc structures 273
Helmholtz energy
– decomposition 347
– minimization 560
– precipitation 318 ff, 337 ff
– spinodal decomposition 417
– vacancies 218
Henry's law 21, 29 ff
Hess' law 660
heterodiffusion 180, 235
heterogeneous nucleation, precipitation 315, 319,
468
heterogeneous regions, interfacial diffusion 484
heterophase fluctuations 414 f, 431 f
heterophases 302
high-angle grain boundaries 485
high damping materials (HIDAMETS) 647
high-pressure generation 666 ff
high-pressure phase transformations 655–695
high-resolution microanalysis 511
high-resolution transmission electron microscopy
(HRTEM) 326, 620
Hillert–Sundman treatment 496 ff
hole hopping, diffusion 205
homogeneity range, phase diagrams 28
homogeneous lattice distortive strain 590
homogeneous second phase precipitation 309–407
homophase fluctuations, spinodal decomposition
414 f, 431 f
Hopf solution 92
Hugoniot elastic limit (HEL) 667
Hume–Rotherey alloys 619 f, 634
hydraulic presses 668 ff
hydrodynamics, spinodal decomposition 446
hydrogen bonds, high-pressure transformations 687 f
hydrostatic pressure 649www.iran-mavad.com
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712. 704 Index
hyperscaling relation, phase transitions 267
hysteresis behavior, martensitic transformations
633 f
ice amorphization 662, 687 ff
ideal solutions
– Henrian 29
– Raoultian 12 f
immiscibility 26
impurities
– diffusion 176 ff, 213
– directional solidification 120
– interfacial diffusion 500
– ionic crystals 231
– metals diffusion 220
– phase transitions 279
– spinodal decomposition 415, 468 f
impurity–vacancy interaction 198, 221
impurity correlation factors 196, 208
incoherent miscibility gaps 570
incommensurate superstructures 273
incubation time, precipitation 319, 343
indirect interstitial mechanism, diffusion 190
indium-based alloys, diffusionless transformations
607, 634
inertial hydrodynamics, spinodal decomposition 446
infrared spectroscopy, high pressure 674
inoculants, solidification 95
inorganic compounds, diffusionless transformations
639
InSb, high-pressure transformations 682
instabilities
– dendritic growth 106, 136 ff
– eutectic growth 157
– solidification 93
integral molar enthalpy 12
interatomic interactions 659
intercalation compounds 205
intercluster spacing 336
interconnected microstructures 377
interconnected structures 364
interdiffusion
– chemical 178 ff
– metals 227
– spinodal decomposition 424, 472
interdiffusion coefficients 233, 357
interfaces
– decomposition 340, 347
– diffusionless transformations 622
– solidification 85 ff, 89 f, 99
interfacial diffusion 481–518
interfacial energies 389 ff
interlamellar spacing 511
intermediate phases 26
– silicon 681
intermetallic phases 26
– ordering 523
– precipitation 314 f
internal friction, diffusionless transformations 587
interphase boundaries 484 ff
interstitial alloys
– atomic ordering 523
– diffusion 224
interstitial ferrous martensites 636
interstitial jumps 213
interstitial mechanism, diffusion 189 f, 216
interstitial solid solutions, diffusion 205
interstitial solutions, phase diagrams 64
interstitials, metal diffusion 231
interstitial-substitutional mechanism, diffusion 191
intrinsic correlation length, solidification 85
intrinsic defects, diffusion 227
intrinsic diffusion 179 f, 184 f
invariants, binary phase diagrams 31
invariant plane strain (IPS) 595, 613 ff
invariant reactions, nomenclature 49
inverse cluster variation method 550
inverse coarsening 380
inverse Monte Carlo technique 550
ionic conductivity, diffusion 183, 189, 204 f, 208 ff
ionic crystals, diffusion 227
ionic solutions, phase diagrams 62
ionic valence, diffusion 181
iron
– alloying elements 611
– massive transformation 500
see also:Fe
iron aluminides, ordering 565
iron-based alloys, martensitic transformations 635
iron whiskers, martensitic transformations 601
irregular structures, eutectic growth 161
irregular tetrahedron approximation, atomic
ordering 553
irreversible thermodynamics, diffusion 181 ff
Ising model
– antiferromagnet 200
– atomic ordering 524
– coarsening 380
– decomposition 367 ff
– ferromagnets 303
– phase transitions 264, 275, 285, 293
– spinodal decomposition 419 ff, 435 ff, 462
isobaric temperature–composition diagrams 51
isopleths, ternary 46
isopropylacrylamide gel 456
isothermal precipitation 315 f
isothermal sections, ternary systems 43, 569 ff
isotope effect, diffusion 196, 213
isotropic pair probabilities, atomic ordering 527
Ivantsov–marginal stability, solidification 118
Ivantsov solution 102 ff, 111, 131
Jänecke coordinates 51
Johnson–Mehl equation 465
jump dicontinuity, interface 89
jump frequency
– decomposition 343
– diffusion 194, 216 f
– metal diffusion 226
jump singularities 247, 262www.iran-mavad.com
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713. Index 705
Kampmann–Wagner approach
– precipitation 382 ff
– spinodal decomposition 433
Kawasaki model 367
kickout mechanism, diffusion 191
Kikuchi natural iteration 525, 554
kinetic coefficients
– dendritic growth 115, 134
– eutectic growth 155
– solidification 87
kinetics
– diffusionless transformations 588, 603
– martensitic transformations 624 f, 632 f
– precipitation 314 f
– spinodal decomposition 416 f
Kirkendall effect 179 f, 233
Kittler–Falikov theory 288
Koch–Cohen cluster 230
Kosterlitz–Thoules phase transitions 268
Kröger–Vink defect notation 228 f
Kurdjumov–Sachs relations 616, 620 f, 637 f
La–Ni–Al system, phase separation 454
lamellar modulations 361 ff
lamellar systems 152, 158
Landau coefficient 127
Landau symmetry classification 246 f
Landau theory
– diffusionless transformations 603
– omega transformations 609
– phase transitions 250 ff, 262 f, 269, 290
– spinodal decomposition 419, 438, 463
Langer–Baron–Miller approximation 359, 423 ff,
440 ff, 471
Langer–Schwartz model
– concomitant processes 382 ff
– spinodal decomposition 433, 445
Langer theory 321 ff
large-amplitude fluctuations 418
large-volume presses 668
laser trace, dendritic growth 101
laser treatment, solidification 96
lasers, high-pressure transformations 667
laths, diffusionless transformations 617
lattice anisotropy, decomposition 420
lattice deformations, diffusionless transformations
620, 639
lattice diffusion length 187
lattice distortive displacements 588
lattice-gas model
– diffusion 201
– spinodal decomposition 428, 444 f
lattice relaxation, ordering 523, 573
lattice softening, diffusionless transformations 597
lattice structures
– atomic ordering 529 ff, 558
– diffusion 194, 223
– order parameters 251 f
– phase transitions 271
see also:structures
lattice wave modulations 592, 597
ledges, structural 488
Legendre polynomials, thermodynamics 34
Legendre transformation 538, 660
length scales, solidification 85
Lennard–Jones interaction 294
lens-shaped two phase regions 20
lenticular martensite 618
lever rule
– binary systems 15
– phase diagrams 41, 248
– spinodal decomposition 417
LiF–NaF system, phase diagram 35, 67
Lifshitz–Slyozov–Wagner model 370 ff, 382 f, 452,
456, 463, 471
Lifshitz invariants 273 ff
Lifshitz point 275
light scattering, precipitation 328, 361
limiting slopes, phase boundaries 66
linearized theory, spinodal decomposition 421 f
Liouville equation 440
liquid–gas transitions 246 ff
liquid–liquid miscibility 26
liquid–solid phase transitions 81–170, 246
liquid compounds, phase diagrams 9
liquidus 39 f, 74, 279
lithium alloys, diffusionless transformations
617, 635
local relaxation, atomic ordering 550
localized nucleation models, martensitic
transformations 629
long-range order (lro) configurations
– atomic ordering 529 ff, 539, 546, 566
– phase transitions 71, 288
low-angle tilt boundaries 485
lutidine–water system, decomposition 455
macroscopic diffusion 175 ff
magnetic exchange energy, atomic ordering 570
magnetic ordering, spinodal decomposition 460
magnetic properties, high-pressure
transformations 659
magnetic solid systems, phase transitions 246
magnetization, precipitates 339
magnetostrictive couplings 271
magnon modes 277
manganese-based alloys, diffusionless transformations
608, 612, 635
Manning model 185 f, 199, 209, 215 f, 226
marginal stability hypothesis, dendritic growth
110, 118
Marqusee–Ross model 373
Marsh–Glicksman model 373
martensitic transformations 260, 587–651
mass action law, ionic crystals 230
mass transfer, diffusion 180
massive transformations, interfacial diffusion 498 ff
mean field approximation (MFA)
– decomposition 359
– phase transitions 255, 259 f, 287 ffwww.iran-mavad.com
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714. 706 Index
– precipitation 318 ff
– spinodal decomposition 417 ff, 431 f
mechanical forces, interface migration 493
mechanical properties, diffusionless
transformations 590
melting
– congruent 29
– phase transitions 283
melting entropy 149
melting point
– atomic ordering 571
– phase diagrams 8, 18
melting temperature
– discontinuous precipitation 505
– solidification 91 ff
mesophases, block-copolymer melts 259, 271
metallic glasses, decomposition 455
metallic substitutional alloys 523 ff
metallic systems
– decomposition 360, 366
– diffusionless transformations 634
– interfacial diffusion 491, 503
– phase separation 322 f
– spinodal decomposition 451 f
– substitutional ordering 523 ff
metals
– dendritic growth 119
– diffusion 220 ff
metastability
– interfacial diffusion 486
– phase transitions 298 ff
– precipitation 316
– spinodal decomposition 417
metastable phase boundaries 39
methanol–cyclohexane mixture, decomposition 457
MgO–CaO system, phase diagram 22
Mg–Zn system 19, 45
microanalytical techniques 326 ff, 511
microclusters 340
microscopic diffusion 189 ff
microscopy, high-pressure transformations 674
microsegregation 131
microstructures
– coarsening 377
– diffusionless transformations 618 ff
– polydispersed 382
– precipitation 314 f
midribs, diffusionless transformations 618 f
migration
– atomic 175
– coarsening 379
– diffusion 186
– interfacial diffusion 484 f, 503 ff, 514
– ionic crystals 228
minerals, diffusionless transformations 639
minimization algorithm
– atomic ordering 525, 554
– Gibbs energy 73
miscibility gap
– atomic ordering 558, 564, 572 f
– binary systems 21
– decomposition 333, 339 ff, 390, 416 f, 460
– phase diagrams 14
– precipitation 314 ff, 401
misfit strain, precipitates 447
mixed alkali effect 209
mixing energies, phase diagrams 9 ff
mobilities
– diffusion 183, 205, 211
– interface migration 485 f, 496, 508
– solidification 89
– spinodal decomposition 421
modified Langer–Schwartz model 382 ff
modulated phases 273
modulated structures, decomposition 360, 364
molar enthalpy, phase diagrams 12
molecular crystal systems 246
molecular dynamics
– interfacial diffusion 484
– phase transitions 293
molecular field approximation 288
molecular field theory 288 ff
monoclinic transformations 600
monodisperse polymers, spinodal decomposition 427
monolayers, absorbed, order-disorder 288
monotectic invariants, phase diagrams 26, 32
Monte Carlo simulations
– atomic ordering 551, 556 f
– decomposition 367
– diffusion 200, 215
– interfacial diffusion 484 ff
– phase transitions 287
– precipitation 321, 396 f
– spinodal decomposition 428 ff, 437 f
morphological instabilities, directional dendritic
growth 139 f
morphology
– decomposition 360 ff
– diffusionless transformations 588, 616, 636
– interfacial diffusion 488
– precipitation 334 ff
– spinodal decomposition 415, 470
Mössbauer spectroscopy (MBS)
– diffusion coefficients 234
– diffusionless transformations 587
– high-pressure transformations 675
mottled microstructure 364
moving boundaries 503
Mullins–Sekerka instability
– dendritic growth 106, 136 ff
– solidification 93, 120, 126 ff
Mullins equation 491 f, 510
mullite, formation 5
multicomponent phase diagrams 14, 39 ff, 47
multicomponent systems, atomic ordering 523
multicritical phenomena, phase transitions 269
multilayers 514
multiple interface transformations, martensitic 633
multisoliton lattice 274
mutual solubility, limited 29 fwww.iran-mavad.com
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715. Index 707
Na–K–F–Cl salt system, phase diagram 49
natrolite frameworks, high pressure 662
nearest neighbors model
– atomic ordering 532, 558
– decomposition 367
– high-pressure transformations 663
– phase diagrams 63, 268, 288, 295
– spinodal decomposition 444 f, 450
needle crystal solution, solidification 101, 113
neighbor interactions 531 f
nematic liquid systems, phase transitions 246
neodynum pentaphosphate, displacive
transformations 600
Nernst–Einstein equation 183 f, 211
neutral stability, directional solidification 126
neutron diffraction, phase transformations 587, 678
Newton–Raphson technique 554
Ni–Al–Mo system, phase separation 381, 450
Ni–Al system
– decomposition 346, 354 f
– interfacial energies 390
Ni–Cu–Al system, precipitation 315 f, 332, 390
Ni–Si system, precipitation 397
Ni–Ti system, displacive transformations 645, 651
nickel-based alloys, displacive transformations 635
Nishiyama–Wassermann relations 616, 620 f, 637 f
nitinol, shape memory effect 642 f
nitrates, displacive transformations 639
nitrides, ordering 523
noble metal-based alloys
– atomic ordering 564
– diffusionless transformations 599 f
noble metals, diffusion 221
nonfaceted systems, eutectic growth 158
nonferrous alloys, diffusionless transformations 637
nonisothermal precipitation 401
nonlinear theories, spinodal decomposition 437 f
nonmetals, diffusionless transformations 638 f
nonstoichiometric compounds, phase diagrams 65
nuclear magnetic resonance (NMR)
– diffusion coefficients 234
– high-pressure transformations 676
nuclear methods, diffusion 231, 234
nucleation
– concomitant processes 381 ff
– decomposition kinetics 332, 340 ff
– eutectic growth 155
– interface migration 495
– martensitic transformations 598, 630
– phase transitions 300
– precipitation 315 ff
– solidification 87 f
– spinodal decomposition 415, 468
numerical modeling, precipitation 381 ff, 399 f
octahedron cluster 535, 573
Olsen–Cohen model 629
omega transformation 609
Onsager coefficients 428
Onsager theorem 181 f, 184, 206
optimization techniques, phase diagrams 34 f
orbital overlaps, high pressure 666
order–disorder transitions
– atomic ordering 567
– high-pressure transformations 664
– kinetics 462
– phase diagrams 71
– solidification 87
– spinodal decomposition 415 f, 460
– symmetry breaking 247 f
order parameters
– interfacial diffusion 486 f
– phase transitions 71, 246 f
– solidification 87
ordering
– atomic 519–581
– pressure-induced 664
– short/long range 68 ff, 529 ff
– spinodal 415
organics, diffusionless transformations 639
orientation relationship, diffusionless
transformations 616
orientational glasses 281 f
oriented clusters, atomic ordering 524
Ornstein–Zernike type susceptibility 264
orthorhombic transformations 600
oscillations, eutectic growth 162
Oseen tensor 440
Ostwald ripening 370
see also:coarsening
outgassing 224
oxidation resistance 565
oxides
– atomic ordering 523
– decomposition 361, 367
oxygen partial pressure, ionic crystals 230
oxygen self-diffusion 178
pair interactions, atomic ordering 527 f, 538
pair variables 527 ff
parabolic directional dentrites 134
parabolic growth, precipitates 351
parabolic Ivantsov solution 104 f, 113
paraboloid symmetry, dendritic growth 115
paramagnetic states, ordered phases 565
Paris–Edinburgh cells 670, 676
partial diffusion coefficients 179 f
partial properties, phase diagrams 11 f
particle (precipitate) coherency 392
particle (precipitate) pinning 498
particle (precipitate) splitting 377
path probability method (PPM) 200, 209
Pb–Sn bicrystals, boundary displacement 506
Péclet number 103 ff, 110, 132
percolation
– diffusion 209
– phase transitions 281
– spinodal decomposition 416, 433, 442 ff
peritectic invariants, phase diagrams 26 f, 31 f
peritectics 163www.iran-mavad.com
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716. 708 Index
peritectoid precipitation 484
perovskites, high pressure 663
pertubations
– dendritic growth 111, 139
– directional solidification 127 f
phase boundaries, phase diagrams 66
phase diagrams 1–80
– CVM/MC methods 554
phase equilibra, atomic ordering 551 ff
phase field models
– directional dendritic growth 151 f
– interfacial diffusion 487
phase rule, Gibbs 7 f
phase separation
– precipitation 314 f, 320 f
– spinodal decomposition 416
phase transformations
– high pressure 655–695
– solidification 85
phase transitions, statistical theories 239–308
phase/microstructure selection, eutectic growth 164
phenomenological theories
– diffusionless transformations 620
– phase transitions 246 ff
– precipitation 398
– spinodal decomposition 416 f
– interfacial diffusion 487
phonons
– diffusionless transformations 597 f
– phase transitions 258, 277, 282, 291
physical correlation factor, diffusion 184, 204
pinning, interfaces 498
piston–cylinder systems 668
pivalic acid (PVA), dendrites 115
plastic crystals
– dendritic growth 116
– phase transitions 258
– solidification 96
plastic deformation, diffusionless transformations
606
plastic response, interface migration 494
plate-shaped martensitic regions 617
point clusters, atomic ordering 524
point defects
– diffusion 193
– phase transitions 280
– spinodal decomposition 415
point variables, atomic ordering 525 f
polarization, phase transitions 246, 258
polydispersity, polymers 427
polygons, atomic ordering 534, 541
polyhedral joints, high-pressure transformations 661
polyhedron, atomic ordering 531 ff
polymer alloys, decomposition 416
polymer mixtures, decomposition 361, 426 f, 455 ff
polymer solutions, phase diagrams 66
polymers, diffusionless transformations 639 f
polymorphic states 611, 640
polynominal representation, excess properties 34
polystyrene–poly(vinyl methyl ether) system,
decomposition 328, 455 ff
polytetrafluoroethylene (PTFE), diffusionless
transformations 640 f
polythermal projections, liquidus surfaces 41
polytypic transitions, shear transformations 613
Porod law 447, 460
positron annihilation diffusionless
transformations 587
potassium, diffusion 232
potential–composition diagrams 5 ff
potential axes, phase diagrams 51
Potts model 255, 259 f, 270 f, 285, 293
Pourbaix diagrams 9 f
power law approximation, precipitation 395 ff
precipitate-growth stages, experimental identification
387
precipitation
– discontinuous 504 f
– homogeneous 309–407
– interfacial diffusion 484 ff
– spinodal decomposition 442
precursor effects
– diffusionless transformations 598 f
– phase transitions 282
predominance diagrams 5 ff
preexponential factor, diffusion 220
preloading, high-pressure transformations 669
premartensitic phase 639
pressure–composition phase diagrams 19
pressure, phase diagrams 6
pressure–temperature diagram, water 54
pressure-driven phase transitions 246, 655–695
pretransformation states, diffusionless 595 f
primary cell spacing, directional dendritic growth
135 f
primary cristallization, phase diagrams 41
primary order parameters, phase transitions 252
pseudo time averages, phase transitions 294
pseudobinary alloys 332
pseudocomponents, phase diagrams 64
pulse transmission/echo method, high-pressure phase
transformations 677
pulsed field gradient NMR, diffusion 234
Purdy–Lange model 506
purity, dendritic growth 116
PVA–ethanol, solidification 115
pyroxene chain silicates, diffusionless transformation
639
quadrupolar glasses, ordering 282
quadrupole moments 252, 258
quantum liquid systems, order parameters 246
quartz, high-pressure phase transformations 661
quasibinary phase diagrams 47
quasichemical model 68
quasieleastic neutron scattering (QNS), diffusion
234
quasiharmonic approximation 292
quasimartensitic transformations 588, 611 ff
quasistationary approximation
– directional dendritic growth 144www.iran-mavad.com
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717. Index 709
– eutectic growth 157
– solidification 92, 125
quenched disorder 279 f
quenched impurities 468 f
quenched-in nuclei 95
quenching
– decomposition 330 f, 339
– diffusionless transformations 587 ff, 643
– precipitation 314 ff
– predecomposition 391
– spinodal decomposition 414 ff, 427
R-phase 639
radioisotopes, diffusion coefficients 232
Raman spectroscopy, high pressure 674
random alloy model 185 ff, 199, 226
random bond, phase transitions 280
random phase approximation 427
random walk theory 189
Rankine–Hugoniot relations 667
Raoultian solutions, ideal 12
rapid diffusion 191
rapid solidification 499, 649
rare-earth alloys, diffusionless transformations
608, 635
rate-determining mechanism, directional
solidification 120
Rayleigh–Benard systems 127
Rayleigh number 124
reaction stoichiometry 5 ff
real-system applications, atomic ordering 561
reciprocal ternary phase diagrams 49
reconstructive phase transitions 260
recrystallization 495, 505
Redlich–Kister form, phase diagrams 34
reduced geometry, phase separation 467
regular solution theory 23
regular structures, eutectic growth 158
relative partial properties, phase diagrams 11 f
relaxation
– atomic ordering 523, 550, 573
– diffusionless transformations 635
– interface migration 495
– spinodal decomposition 424, 467
relaxation methods
– diffusion coefficients 233
– interfacial diffusion 484
– metal diffusion 224
renormalization group methods
– phase transitions 266 ff, 271, 279
– spinodal decomposition 419 f, 429 f
reptation, spinodal decomposition 428
residual acitivity, diffusion coefficients 232
resistivity, electrical, diffusionless transformations
587
retarding forces, interface migration 492
Reynolds number 124
Rigsbee–Aaronson model 488
Rikvold–Gunton model 398
rotational jumps 197
rotational symmetry, dendritic growth 115
roughening transition 268
Rouse model 428
Saffmann–Taylor limit 130
salt system, reciprocal ternary 49 f
salts, transparent, solidification 115
scaling
– dendritic growth 110 f, 114, 135
– directional solidification 124
– interfacial diffusion 502
– solidification 94 f
– spinodal decomposition 425, 430, 445 f, 453 f
scaling laws
– eutectic growth 156
– phase transitions 260 f, 266, 277
– precipitation 395 ff
scanning transmission electron microscope (STEM),
microanalysis 511
scattering data, phase transitions 250, 260, 287
scattering techniques, precipitates 328 f, 423
Schottky defects 227
Schreinemaker rule 46, 57
second-neighbor interactions 535, 543 ff, 557 f
second-order transitions 249, 252, 258, 260 f
second-phase precipitation, homogeneous 309–407
secondary ion mass spectroscopy (SIMS), diffusion
232
segregation, interfacial 484, 489 f
segregation coefficients 125
self-accomodation, diffusionless transformations
619, 636
self-averaging, phase transitions 280
self-diffusion 176 ff
– grain boundaries 491
– isotope effect 215
– metals 202
self-similarity, precipitation 395 ff
semiconductors
– atomic ordering 523
– dendritic growth 119
– zincblende-type, high pressure 679
shallow quenching 427
shape deformations, diffusionless
transformations 600 f, 615
shape-memory alloys (SMA) 590 ff, 633 ff, 638,
641 f
sharp interface model 340
shear transformations 606, 613, 636
shock wave techniques, high-pressure transformations
667
Shoji–Buerger lattice deformations 594
short-circuit diffusion 186 ff
short-range order (sro) configurations
– atomic ordering 529 ff, 539
– phase transitions 68, 288
short-wavelength distortions, phase transitions 283
shuffles, diffusionless transformations
588 ff, 596 ff, 606 ff
Si–O–Si angles 661www.iran-mavad.com
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718. 710 Index
SiC–AlN ceramics, decomposition 454
side branching, dendrites 108 ff, 143, 151
silica, phase transformations 639, 661
silicate slags 69
silicon, high-pressure phase transformations 680
silicon-based alloys, rapid solidification 499
silver alloys, diffusionless transformations 634
simplex, atomic ordering 534
simulation techniques, phase transitions 293
single chain dynamics 428
single-phase solid solutions 15
single sites, phase transitions 289
singularities, critical 260
SiO
2
–MgO–MnO system, phase diagram 74
site cluster approximation 560
site occupation operator 526
slip, diffusionless transformations 590, 605, 622
slow growth rates, solidification 96
slow mode theory 428
slowing down, phase transitions 277
small-amplitude fluctuations, spinodal decomposition
418
small-angle neutron scattering (SANS), precipitates
328, 391
small-angle X-ray scattering (SAXS), precipitates
328, 360
Snoek effect, diffusion 233
snow flakes, dendritic growth 100
soft phonons 277, 282, 291
softmode phase transitions
– diffusionless 596
– martensitic transformations 631
solid compounds, phase diagrams 9
solid binary mixture systems 246
solid–liquid equilibra 18
solid–liquid interface, directional solidification 120
solid–liquid miscibility, phase diagrams 14
solid–solid transformations 484, 490
solid solutions
– decomposition 314 f, 385
– diffusionless transformations 635
– phase diagrams 15
solid-state diffusion 189
solidification 81–170
solidus line 279
soliton staircase 274
solubility
– mutual 29
– precipitation 316, 370, 393
solutal dendrites 117
solute–solute binding 225
solute–vacancy binding 222
solute drag, interfacial motion 496
solute enhancements factors 221
solution models, phase diagrams 62 ff
solvability theory, dendritic growth 106, 116
solvent enhancement factors, diffusion 221
solvus lines 23, 314 f
Soret effect 181
sound velocity, high-temperature transformations 677
space groups, atomic ordering 526 f575
space model, ternary 39
spacing
– dendritic growth 111, 117
– directional solidification 130 ff
– eutectic growth 159
– interfacial diffusion 485, 509
– solidification 99
– spinodal decomposition 426, 450
see also:lattice spacing, interface spacing,
cell spacing
spectroscopic techniques, high-temperature
transformations 674
sphere crystal growth 92
spin glass 281
spin representation, Ising 286
spin variables 524, 570
spinodal alloys 363 ff
spinodal decomposition 332, 354 ff, 409–480
– precipitation 319 ff
spinodal point, phase transitions 22, 299
spins, phase transitions 255, 259 f
split sphere anvil technology 670
splitting
– clusters 352 f
– coarsening 377 f
– spinodal decomposition 433
sponge-like microstructure 364
spontaneous growth, ordered domains 462
spontaneous magnetization 246
stability
– dendritic growth 118
– nucleation 350
– spinodal decomposition 417
stability domains, phase diagrams 8
stability length
– dendritic growth 106, 111
– solidification 93
stability limit, phase transitions 254
stabilizing, diffusionless transformations 611, 625
stable pair, exchange reaction 63
stacking faults, diffusionless transformations 601, 634
stacking sequences, diffusionless transformations
595, 613
staggered fields, ordering 248
standard model, directional solidification 125
standard molar Gibbs energy 6
standard states, phase diagrams 11 f
static pressure devices 668
statistical theories, phase transitions 239–308
steady cooperative growth 507
steady-state nucleation rate 342
steels, precipitation 366
steepest descent technique, atomic ordering 554
stiffness tensor 447
Stirling approximation
– atomic ordering 552
– diffusion 219
– phase diagrams 13
stoichiometric compositions 566
stoichiometric crystals 227
stoichiometric product phase 515www.iran-mavad.com
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719. Index 711
stoichiometry, phase diagrams 5 ff, 28
Stokes' law, cluster diffusion 456
strain
– diffusionless transformations 588, 606, 647 f
– interface migration 493
– precipitation 324, 337
strain spinodals 631
strengthening, martensite 649
stress–temperature diagramm 645
stress induced transformations
– martensitic 633
– diffusionless 647
stress relaxation, diffusionless transformations 635
strontium titanate, displacive transformations
591, 607
structural distortions, high-pressure transformations
659, 665 f
structural models 5
structure functions, spinodal decomposition 442
structure modulation 360, 364
structures
– diffusionless transformations 593 f
– eutectic growth 158 f
– grain boundaries 484 f
– ordered 523 f
– precipitation 314 f
styrene–butadiene mixture, phase separation 457
subcluster probabilities, atomic ordering 529
subgrains, stored energy 495
sublattice magnetization 246
sublattices
– atomic ordering 529
– interfacial diffusion 484
– phase transitions 62, 251
– spinodal decomposition 415
substitutional alloys
– atomic ordering 523 ff
– diffusion 221, 226
substitutional ferrous martensite 636
substitutional impurities, spinodal decomposition
415
substitutional solution model, ideal 12
substructures, diffusionless transformations 616, 637
succinonitrile (SCN)
– dendritic growth 109, 114 f, 139 f
– solidification 96 ff
sulfides, diffusionless transformations 639
superconductivity, pressure dependence 666
supercooling
– dendritic growth 109, 131, 135
– eutectic growth 154
– solidification 88 ff, 92
supercritical clusters 434
superelastic alloys 650
superhard materials, high-pressure transformations
659
superheated melts 95
supersaturated matrix, precipitation 350
supersaturation
– coarsening 377
– decomposition 330, 345, 359, 414 f
– diffusionless transformations 635
– interfacial diffusion 499
– precipitation 314 f, 323 ff, 396 ff
– spinodal decomposition 414 f
superstructures
– atomic ordering 529, 546
– b.c.c. lattice 577
– f.c.c. lattice 575
– phase transitions 271 ff
surface diffusion 180 f, 192
– coefficient measurements 234
surface effects
– disorder 283
– phase transitions 279 f
– spinodal decomposition 467 f
surface melting 283
surface tension 88, 124 ff
surfactant micellar solutions 456
symmetry, dendritic growth 115
symmetry breaking, order parameters 247
symmetry lowering, ferroic phase transitions 608
symmetry properties, phase transitions 256, 270
synchroton radiation, high-pressure studies 667, 670
syntectic invariants, phase diagrams 32
tail instability, solidification 137
tangent constructions 11 f, 16
temperature–composition phase diagrams 5 ff, 14
temperature–concentration plane, spinodal
decomposition 435
temperature–stress diagram, diffusionless
transformations 647
temperature gradients
– atomic ordering 551 ff
– diffusion 181
– spinodal decomposition 470
temperature ranges
– decomposition 340
– diffusionless transformations 587, 590 f, 607
– interfacial diffusion 501
ternary alloys
– pair variables 528
– phase transitions 285
– precipitation 334, 397
ternary mixtures, spinodal decomposition 415
ternary systems 5, 39 ff
– atomic ordering 569 f
tetrahedral framework, high-pressure
transformations 661
tetrahedron, atomic ordering 573
tetrahedron approximation, phase transitions 288
tetrahedron clusters, atomic ordering 536 f, 544 f
tetrahedron–octahedron approximation 553 ff, 560
thallium alloys, diffusionless transformations 634
thermal dendrites 116
thermal diffusion
– dendritic growth 108
– solidification 91
thermal fluctuations
– dendritic growth 113www.iran-mavad.com
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720. 712 Index
– solidification 87
– spinodal decomposition 414 f
thermal gradients, directional solidification 126
thermally induced martensitic transformations 633
thermodynamic compilations, phase diagrams 77
thermodynamic factor
– atomic ordering 565
– diffusion 205
thermodynamic variables, conjugate 247
thermodynamics 1–80
– diffusionless transformations 603 ff
– irreversible 181 f
– martensitic transformations 624 f
– phase separation 322
– precipitation 317 f
– spinodal decomposition 416 f
– two-component systems 121
thermoelastic behavior, martensitic
transformations 622, 632 f
thin layer methods, diffusion coefficients 231
three phase stability 9
Ti–Al–X systems, atomic ordering 562
Ti–Ni–Cu alloys, martensite 631
tie lines 15, 41, 60
tilt grain boundaries 485 ff, 495 f
tilted domains 156
tilted lamellar arrays 151
time-dependent nucleation rate 343
time residence algorithms, Monte Carlo simulations
485
tip radius
– dendritic growth 110 ff, 116 f, 120
– directional dendritic growth 133, 143
titanium alloys, diffusionless transformations
607, 617, 623, 634 ff
tomographic atom probe (TAP), precipitates 328
topology
– phase diagrams 45
– phase transitions 25, 296
torques
– discontinuous precipitation 506
– interfacial diffusion 499
trace impurities, interfacial diffusion 500
tracer correlation factor
– defective systems 204
– microscopic diffusion 193
tracer diffusion 178 ff, 182 ff
– interfacial 490
– metals 226
– methods 231
training, shape memory effect 643
transformation hysteresis, martensitic 633 f
transformation-induced plasticity (TRIP) 593, 647
transformations, interfacial diffusion 481–518
transmission electron micrographs
– decomposition 360, 366
– diffusionless transformations 600
– directional dendritic growth 149
transparent salts, dendritic growth 115
tricritical point
– atomic ordering 565
– phase transitions 269, 272
– spinodal decomposition 460
triple points, phase diagrams 8, 55
true phase diagram sections 56 f
trunk spacing, dendrites 144 ff
twinning 595, 600, 605, 615, 618, 622, 634
twist boundaries 491, 495
two-component systems, directional solidification 121
two-phase stability 9
two-phase regions 20
two sided model, dendritic growth 102
two-way shape memory effect, diffusionless
transformations 644
ultrasonic sound velocity, high-pressure
transformations 677
undercooling
– dendritic growth 118, 146
– diffusionless transformations 612
– martensitic transformations 625
– precipitation 325
– solidification 95 ff
undistorted lines, diffusionless transformations 592
univariant lines 41
univariant phase regions 31
universality principle
– phase transitions 266, 293
– spinodal decomposition 431 f
unstable states, spinodal decomposition 417
uphill diffusion 357, 422
vacancies
– interfacial diffusion 485
– spinodal decomposition 468 f
vacancy–impurity complexes, diffusion 223
vacancy concentration
– diffusion 218
– ionic crystals 230
vacancy mechanism, diffusion 191
vacancy-wind effect 182 f, 185 f, 208 ff
van der Waals forces 24, 255
vapor–liquid equilibrium 18
variance, phase diagrams 31
vector model, order parameters 251
vibrational displacements 588, 597
vibrational energy, atomic ordering 573
vibrational modes, high-pressure transformations 674
viscoelastic phase separation 470
viscous hydrodynamics, phase separation 446
Volmer–Weber theory 342 ff
volume fractions
– diffusionless transformations 587
– eutectic growth 158
– precipitation 373 f, 395
– spinodal decomposition 427
volume relaxation, atomic ordering 550
volumetric pressure techniques, phase
transformations 674www.iran-mavad.com
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721. Index 713
water
– high-pressure transformations 687 ff
– phase diagram 54, 688
wave modulations, diffusionless transformations 592
weak fluctuations, spinodal decomposition 418 ff
wetting phase transitions 283
whiskers, martensitic transformations 601
Widmanstätten precipitates 507
Willis cluster 230
wollastonite, diffusionless transformations 639
Wulff construction
– decomposition 344 f
– interfacial diffusion 488
– solidification 90
wurtzite
– diffusionless transformations 639
– high-pressure transformations 680, 688
X-ray absorption spectroscopy (XAS), transformations
676
X-ray diffraction, phase transformations 587, 612,
678
X-ray microanalysis 326, 511
Young's modulus, diffusionless transformations 590
Zeldovich factor 341 ff
zeolite frameworks, high-pressure transformations
662
zero-phase fraction (ZPF) lines 58
zincblende type semiconductors, high-pressure
transformations 679
zirconium, diffusionless transformations 598 ff, 639
zirconium alloys, diffusionless transformations
607, 617, 634
zone melting 98
Zr–Ti–Cu–Ni–Be glass, phase separation 455www.iran-mavad.com
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